Recent zbMATH articles in MSC 35Qhttps://zbmath.org/atom/cc/35Q2024-11-01T15:51:55.949586ZUnknown authorWerkzeugPinning in the extended Lugiato-Lefever equationhttps://zbmath.org/1544.340342024-11-01T15:51:55.949586Z"Bengel, Lukas"https://zbmath.org/authors/?q=ai:bengel.lukas"Pelinovsky, Dmitry"https://zbmath.org/authors/?q=ai:pelinovsky.dmitry-e"Reichel, Wolfgang"https://zbmath.org/authors/?q=ai:reichel.wolfgangSummary: We consider a variant of the Lugiato-Lefever equation (LLE), which is a nonlinear Schrödinger equation on a one-dimensional torus with forcing and damping, to which we add a first-order derivative term with a potential \(\varepsilon V(x)\). The potential breaks the translation invariance of LLE. Depending on the existence of zeroes of the effective potential \(V_{\mathrm{eff}}\), which is a suitably weighted and integrated version of \(V\), we show that stationary solutions from \(\varepsilon=0\) can be continued locally into the range \(\varepsilon \not=0\). Moreover, the extremal points of the \(\varepsilon\)-continued solutions are located near zeros of \(V_{\mathrm{eff}}\). We therefore call this phenomenon \textit{pinning} of stationary solutions. If we assume additionally that the starting stationary solution at \(\varepsilon=0\) is spectrally stable with the simple zero eigenvalue due to translation invariance being the only eigenvalue on the imaginary axis, we can prove asymptotic stability or instability of its \(\varepsilon\)-continuation depending on the sign of \(V_{\mathrm{eff}}^{\prime}\) at the zero of \(V_{\mathrm{eff}}\) and the sign of \(\varepsilon\). The variant of the LLE arises in the description of optical frequency combs in a Kerr nonlinear ring-shaped microresonator which is pumped by two different continuous monochromatic light sources of different frequencies and different powers. Our analytical findings are illustrated by numerical simulations.Almost sure scattering for the one dimensional nonlinear Schrödinger equationhttps://zbmath.org/1544.350012024-11-01T15:51:55.949586Z"Burq, Nicolas"https://zbmath.org/authors/?q=ai:burq.nicolas"Thomann, Laurent"https://zbmath.org/authors/?q=ai:thomann.laurentIn this book, the authors study the long time dynamics for the solutions to the Cauchy problem for the one-dimensional nonlinear Schrödinger equation \(i\partial_sU+\partial_y^2U=|U|^{p-1}U\), where \((s,y)\in\mathbb{R}\times\mathbb{R},\;p>1\), and \(U(s_0,y)=U_0\) is a random initial condition with low Sobolev regularity. As we know, on compact manifolds, many probability measures are invariant by the flow of the linear Schrödinger equation, such as Wiener measures, and it is sometimes possible to modify them suitably and get invariant (e.g. Gibbs measures) or quasi-invariant measures for the nonlinear problem. While on Euclidean space \(\mathbb{R}^d\), the large time dispersion shows that the only invariant measure is the \(\delta\)-measure on the trivial solution \(u=0\), and the good notion to track is whether the nonlinear evolution of the initial measure is well described by the linear evolution. In this work, the authors achieve this conclusion. More precisely, the authors first define measures on the space of initial data for which they can describe precisely the nontrivial evolution by the linear Schrödinger flow. Second, they prove that the nonlinear evolution of these measures is absolutely continuous with respect to their linear evolutions. Actually, they give precise and optimal bounds on the Radon-Nikodym derivatives of these measures with respect to each other and characterise their \(L^p\) regularity. And then, they get benefit from this precise description to prove the global well-posedness for \(p>1\) and almost sure scattering for \(p>3\). This is the first occurrence where the description of quasi-invariant measures allows to get quantitative asymptotics for the nonlinear evolution.
Reviewer: Jiqiang Zheng (Beijing)Symmetry problems. The Navier-Stokes problemhttps://zbmath.org/1544.350052024-11-01T15:51:55.949586Z"Ramm, Alexander G."https://zbmath.org/authors/?q=ai:ramm.alexander-gSee the review of the original edition in [Zbl 1412.35003].The Navier-Stokes problemhttps://zbmath.org/1544.350062024-11-01T15:51:55.949586Z"Ramm, Alexander G."https://zbmath.org/authors/?q=ai:ramm.alexander-gSee the review of the original edition in [Zbl 1461.35003].Uniqueness of lump solution to the KP-I equationhttps://zbmath.org/1544.350122024-11-01T15:51:55.949586Z"Liu, Yong"https://zbmath.org/authors/?q=ai:liu.yong.1"Wei, Juncheng"https://zbmath.org/authors/?q=ai:wei.juncheng"Yang, Wen"https://zbmath.org/authors/?q=ai:yang.wenSummary: The KP-I equation has family of solutions which decay to zero at space infinity. One of these solutions is the classical lump solution, which is a traveling wave, and the KP-I equation in this case reduces to the Boussinesq equation. In this paper we classify all the `lump-type' solutions of the Boussinesq equation. Using a robust inverse scattering transform developed by Bilman-Miller for the Schrödinger equation, we show that the lump-type solutions are rational and their \(\tau\) functions have to be polynomials of degree \(k(k+1)\) for some integer \(k\). In particular, this implies that the lump solution is the unique ground state of the KP-I equation (as conjectured by Klein-Saut). The problem studied in this paper was mentioned in Airault-McKean-Moser, our result can be regarded as a two-dimensional analogy of their theorem on the classification of rational solutions for the KdV equation.
{\copyright} 2024 The Author(s). The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.New asymptotic lower bound for the radius of analyticity of solutions to nonlinear Schrödinger equationhttps://zbmath.org/1544.350142024-11-01T15:51:55.949586Z"Getachew, Tegegne"https://zbmath.org/authors/?q=ai:getachew.tegegne"Belayneh, Birilew"https://zbmath.org/authors/?q=ai:belayneh.birilewSummary: In this paper, we show that the radius of analyticity \(\sigma (t)\) of solutions to the one-dimensional nonlinear Schrödinger (NLS) equation
\[
i \partial_t u + \partial_x^2u=|u|^{p - 1}u
\]
is bounded from below by \(c|t|^{- \frac{2}{3}}\) when \(p>3\) and by \(c|t|^{- \frac{4}{5}}\) when \(p=3\) as \(|t| \to +\infty\), given initial data that is analytic with fixed radius. This improves results obtained by
\textit{A. Tesfahun} [J. Differ. Equations 263, No. 11, 7496--7512 (2017; Zbl 1375.35436)]
for \(p=3\) and
\textit{J. Ahn} et al. [Discrete Contin. Dyn. Syst. 40, No. 1, 423--439 (2020; Zbl 1516.35204)]
for any odd integers \(p>3\), where they obtained a decay rate \(\sigma (t) \geq c|t|^{-1}\) for larger \(t\). The proof of our main theorems is based on a modified Gevrey space introduced in
[\textit{T. T. Dufera} et al., J. Math. Anal. Appl. 509, No. 2, Article ID 126001, 13 p. (2022; Zbl 1510.35329)],
the local smoothing effect, maximal function estimate of the Schrödinger propagator, a method of almost conservation law, Schrödinger admissibility and one-dimensional Sobolev embedding.New analytical wave structures of the \((3+1)\)-dimensional extended modified Ito equation of seventh-orderhttps://zbmath.org/1544.350162024-11-01T15:51:55.949586Z"Şenol, Mehmet"https://zbmath.org/authors/?q=ai:senol.mehmet-burak"Gençyiğit, Mehmet"https://zbmath.org/authors/?q=ai:gencyigit.mehmet"Demirbilek, Ulviye"https://zbmath.org/authors/?q=ai:demirbilek.ulviye"Akinyemi, Lanre"https://zbmath.org/authors/?q=ai:akinyemi.lanre"Rezazadeh, Hadi"https://zbmath.org/authors/?q=ai:rezazadeh.hadi(no abstract)Lie symmetry analysis, particular solutions and conservation laws for the dissipative (2+1)-dimensional AKNS equationhttps://zbmath.org/1544.350182024-11-01T15:51:55.949586Z"Tao, Sixing"https://zbmath.org/authors/?q=ai:tao.sixing(no abstract)Lie symmetry analysis of seventh order Caudrey-Dodd-Gibbon equationhttps://zbmath.org/1544.350242024-11-01T15:51:55.949586Z"Sharma, Hariom"https://zbmath.org/authors/?q=ai:sharma.hariom"Arora, Rajan"https://zbmath.org/authors/?q=ai:arora.rajanSummary: In the present paper, seventh order Caudrey-Dodd-Gibbon (CDG) equation is solved by Lie symmetry analysis. All the geometry vector fields of seventh order KdV equations are presented. Using Lie transformation seventh order CDG equation is reduced into ordinary differential equations. These ODEs are solved by power series method to obtain exact solution. The convergence of the power series is also discussed.A Lipschitz metric for \(\alpha\)-dissipative solutions to the Hunter-Saxton equationhttps://zbmath.org/1544.350322024-11-01T15:51:55.949586Z"Grunert, Katrin"https://zbmath.org/authors/?q=ai:grunert.katrin"Tandy, Matthew"https://zbmath.org/authors/?q=ai:tandy.matthewSummary: We explore the Lipschitz stability of solutions to the Hunter-Saxton equation with respect to the initial data. In particular, we study the stability of \(\alpha\)-dissipative solutions constructed using a generalised method of characteristics approach, where \(\alpha\) is a function determining the energy loss at each position in space.Instability of \(H^1\)-stable periodic peakons for the \(\mu\)-Camassa-Holm equationhttps://zbmath.org/1544.350362024-11-01T15:51:55.949586Z"Deng, Xijun"https://zbmath.org/authors/?q=ai:deng.xijun"Chen, Aiyong"https://zbmath.org/authors/?q=ai:chen.aiyongIn the paper under review, the authors consider the instability of periodic peakons of the \(\mu\)-Camassa-Holm (\(\mu\)-CH) equation
\[
\mu(u_t)-u_{txx}=-2\mu(u)u_x+2u_xu_{xx}+uu_{xxx},\quad t>0,\quad x\in \mathbb{S}^1,
\]
where \(\mu(u)\) is the mean of \(u\).
The main result is given in Theorem 2, where it is shown the existence of a function \(u_0\in H^1_{per}(\mathbb{S}^1)\cap W^{1,\infty}(\mathbb{S}^1)\), sufficiently close to the periodic peakon solution of the \(\mu\)-CH equation, but whose derivative of the corresponding solution of the equation subject to the initial datum \(u_0\) is not close to the derivative of the peakon solution.
Reviewer: Igor Leite Freire (São Carlos)Transversal spectral instability of periodic traveling waves for the generalized Zakharov-Kuznetsov equationhttps://zbmath.org/1544.350372024-11-01T15:51:55.949586Z"Natali, Fábio"https://zbmath.org/authors/?q=ai:natali.fabio-m-amorinSummary: In this paper, we determine the transversal instability of periodic traveling wave solutions of the generalized Zakharov-Kuznetsov equation in two space dimensions. Using an adaptation of the arguments in [\textit{F. Rousset} and \textit{N. Tzvetkov}, Math. Res. Lett. 17, No. 1, 157--169 (2010; Zbl 1222.35028)] in the periodic context, it is possible to prove that all positive and one-dimensional \(L\)-periodic waves are spectrally (transversally) unstable. In addition, when periodic waves that change their sign exist, we also obtain the same property when the associated projection operator defined in the zero mean Sobolev space has only one negative eigenvalue.Optimal decay estimate and asymptotic profile for solutions to the generalized Zakharov-Kuznetsov-Burgers equation in 2Dhttps://zbmath.org/1544.350462024-11-01T15:51:55.949586Z"Fukuda, Ikki"https://zbmath.org/authors/?q=ai:fukuda.ikki"Hirayama, Hiroyuki"https://zbmath.org/authors/?q=ai:hirayama.hiroyukiSummary: We consider the Cauchy problem for the generalized Zakharov-Kuznetsov-Burgers equation in 2D. This is one of the nonlinear dispersive-dissipative equations, which has a spatial anisotropic dissipative term \(-\mu u_{xx}\). In this paper, we prove that the solution to this problem decays at the rate of \(t^{-\frac{3}{4}}\) in the \(L^\infty\)-sense, provided that the initial data \(u_0(x, y)\) satisfies \(u_0\in L^1(\mathbb{R}^2)\) and some appropriate regularity assumptions. Moreover, we investigate the more detailed large time behavior and obtain a lower bound of the \(L^\infty\)-norm of the solution. As a result, we prove that the given decay rate \(t^{-\frac{3}{4}}\) of the solution to be optimal. Furthermore, combining the techniques used for the parabolic equations and for the Schrödinger equation, we derive the explicit asymptotic profile for the solution.Three examples of residual pathologieshttps://zbmath.org/1544.350692024-11-01T15:51:55.949586Z"Ghisi, Marina"https://zbmath.org/authors/?q=ai:ghisi.marina"Gobbino, Massimo"https://zbmath.org/authors/?q=ai:gobbino.massimoAuthors' abstract: Several counterexamples in analysis show the existence of some special object with some sort of pathological behavior. We present three different examples where the pathological behavior is not an isolated exception, but it is the ``typical'' behavior of the ``generic'' object in a suitable class, where here generic means residual in the sense of Baire category. The first example is the revisitation of a classical result concerning approximate differentiation. The second example is the derivative loss for solutions to linear wave equations with time-dependent Hölder continuous propagation speed. The third result is the derivative loss for solutions to transport equations with non-Lipschitz velocity field.
Reviewer: Giovanni S. Alberti (Genova)Bifurcation, stability, and nonlinear parametric effects on the solitary wave profile of the Riemann wave equationhttps://zbmath.org/1544.350742024-11-01T15:51:55.949586Z"Khan, Kamruzzaman"https://zbmath.org/authors/?q=ai:khan.kamruzzaman"Islam, Md. Ekramul"https://zbmath.org/authors/?q=ai:islam.md-ekramul"Akbar, M. Ali"https://zbmath.org/authors/?q=ai:ali-akbar.mSummary: The enduring stability exhibited by solitons is strikingly demonstrated as a soliton pulse traverses an ideal lossless optical fibre, thereby highlighting a compelling attribute for their integration into optical communication systems. In this study, we employ the improved Bernoulli sub-equation function method to systematically derive stable and functionally robust soliton solutions for the Riemann wave equation. The stability of the soliton solutions is demonstrated through their composition involving hyperbolic and exponential functions, among others. The physical significance of these solutions is meticulously analyzed by presenting 2D and 3D graphs that illustrate the behaviour of the solutions for specific parameter values. Additionally, a comprehensive investigation into the influence of the nonlinear parameter on the wave velocity and solution curve is conducted. The study further explores local stability through bifurcation and phase plane analysis. Our findings affirm the reliability of the improved Bernoulli sub-equation function method and suggest its potential application in future endeavours to uncover diverse and novel soliton solutions for other nonlinear evolution equations encountered in the realms of mathematical physics and engineering.Instability of periodic waves for the Korteweg-de Vries-Burgers equation with monostable sourcehttps://zbmath.org/1544.350762024-11-01T15:51:55.949586Z"Folino, Raffaele"https://zbmath.org/authors/?q=ai:folino.raffaele"Naumkina, Anna"https://zbmath.org/authors/?q=ai:naumkina.anna"Plaza, Ramón G."https://zbmath.org/authors/?q=ai:plaza.ramon-gSummary: In this paper, it is proved that the KdV-Burgers equation with a monostable source term of Fisher-KPP type has small-amplitude periodic traveling wave solutions with finite fundamental period. These solutions emerge from a subcritical local Hopf bifurcation around a critical value of the wave speed. Moreover, it is shown that these periodic waves are spectrally unstable as solutions to the PDE, that is, the Floquet (continuous) spectrum of the linearization around each periodic wave intersects the unstable half plane of complex values with positive real part. To that end, classical perturbation theory for linear operators is applied in order to prove that the spectrum of the linearized operator around the wave can be approximated by that of a constant coefficient operator around the zero solution, which intersects the unstable complex half plane.Nonstandard solutions for a perturbed nonlinear Schrödinger system with small coupling coefficientshttps://zbmath.org/1544.350882024-11-01T15:51:55.949586Z"An, Xiaoming"https://zbmath.org/authors/?q=ai:an.xiaoming"Wang, Chunhua"https://zbmath.org/authors/?q=ai:wang.chunhua.1This paper deals with ``the following weakly coupled nonlinear Schrödinger system
\[
\left\{ \begin{array}{ll} -\epsilon^2\Delta u_1 + V_1(x)u_1 = |u_1|^{2p-2}u_1 + \beta |u_1|^{p-2}|u_2|^pu_1, & x\in\mathbb{R}^N \\
-\epsilon^2\Delta u_2 + V_2(x)u_2 = |u_2|^{2p-2}u_2 + \beta |u_2|^{p-2}|u_1|^pu_2, & x\in\mathbb{R}^N \end{array} \right.
\]
where \(\epsilon>0\), \(\beta\in\mathbb{R}\) is a coupling constant, \(2p\in (2,2^*)\) with \(2^* = \frac{2N}{N - 2}\) if \(N\geq 3\) and \(+\infty\) if \(N = 1,2\), \(V_1\) and \(V_2\) belong to \(C(\mathbb{R}^N,[0,\infty))\).''
When \(p\ge 2\) and \(|\beta|<\beta_0\) for some positive constant \(\beta_0>0\), By using variational methods and the penalized technique, the paper shows that the problem has a family of nonstandard solutions \(\{w_{\epsilon} = (u^1_{\epsilon},u^2_{\epsilon}):0<\epsilon<\epsilon_{0}\}\) concentrating synchronously at the common local minimum of \(V_1\) and \(V_2\). All decay rates of \(V_i(i=1,2)\) especiallly the compactly supported case are included. Due to the loss of monotonicity of higher energy, the peak is located by the local Pohozaev identity instead of the usual energy comparison. Moreover, a type of concentration-compactness principle in weakly coupled nonlinear Schrödinger systems is established to ensure the nontrivial properties of both sectors of \(w_\epsilon\) (\(\lim_{\varepsilon>0}\|u^i_\epsilon\|_{L^{\infty}(\mathbb{R}^N)}>0,i=1,2\)).
Reviewer: Shuangjie Peng (Wuhan)On the boundary control problem associated with a fourth order parabolic equation in a two-dimensional domainhttps://zbmath.org/1544.350972024-11-01T15:51:55.949586Z"Dekhkonov, Farrukh"https://zbmath.org/authors/?q=ai:dekhkonov.farrukh-n"Li, Wenke"https://zbmath.org/authors/?q=ai:li.wenkeSummary: In this paper, we consider a boundary control problem associated with a fourth order parabolic type equation in a bounded two-dimensional domain. The solution with the control function on the border of the considered domain is given. The constraints on the control are determined to ensure that the average value of the solution within the considered domain attains a given value. The initial-boundary problem is solved by the Fourier method, and the control problem under consideration is analyzed with the Volterra integral equation. The existence of admissible control is proved by the Laplace transform method.Turing instability and amplitude equation of reaction-diffusion system with multivariablehttps://zbmath.org/1544.351032024-11-01T15:51:55.949586Z"Zheng, Qianqian"https://zbmath.org/authors/?q=ai:zheng.qianqian"Shen, Jianwei"https://zbmath.org/authors/?q=ai:shen.jianweiSummary: In this paper, we investigate pattern dynamics with multivariable by using the method of matrix analysis and obtain a condition under which the system loses stability and Turing bifurcation occurs. In addition, we also derive the amplitude equation with multivariable. This is an effective tool to investigate multivariate pattern dynamics. The example and simulation used in this paper validate our theoretical results. The method presented is a novel approach to the investigation of specific real systems based on the model developed in this paper.Uniform-in-mass global existence for 4D Dirac-Klein-Gordon equationshttps://zbmath.org/1544.351072024-11-01T15:51:55.949586Z"Zhao, Jingya"https://zbmath.org/authors/?q=ai:zhao.jingyaSummary: We are interested in four-dimensional Dirac-Klein-Gordon equations, a fundamental model in particle physics. The main goal of this paper is to establish global existence of solutions to the coupled system and to explore their long-time behavior. The results are valid uniformly for mass parameters varying in the interval \([0, 1]\).N-soliton solutions for the three-component Dirac-Manakov system via Riemann-Hilbert approachhttps://zbmath.org/1544.351142024-11-01T15:51:55.949586Z"Wang, Yuxia"https://zbmath.org/authors/?q=ai:wang.yuxia"Huang, Lin"https://zbmath.org/authors/?q=ai:huang.lin.3|huang.lin|huang.lin.1"Yu, Jing"https://zbmath.org/authors/?q=ai:yu.jing.1|yu.jingSummary: This investigation delves into the application of the well-established Riemann-Hilbert method for the elucidation of the N-solitons solution of the three-component Dirac-Manakov system. The analytical process is structured in two fundamental steps. Initially, the inverse scattering method is employed to establish a pivotal connection between the solution of the three-component DiracManakov system and the associated Riemann-Hilbert problem. Subsequently, we systematically address the resolution of this pertinent Riemann-Hilbert problem. Through the assignment of specific values to the relevant parameters in our solution, we adeptly generate graphical representations that vividly illustrate the nuanced dynamics inherent in the solution of the three-component DiracManakov system.Existence of strong solutions for a compressible fluid-solid interaction system with Navier slip boundary conditionshttps://zbmath.org/1544.351152024-11-01T15:51:55.949586Z"Djebour, Imene Aicha"https://zbmath.org/authors/?q=ai:djebour.imene-aichaSummary: We consider a fluid-structure interaction system coupling a viscous fluid governed by the compressible Navier-Stokes equations and a rigid body immersed in the fluid and modeled by the Newton's law. In this work, we consider the Navier slip boundary conditions. Our aim is to show the local in time existence and uniqueness of the strong solution to the corresponding problem. The main step of this work is that we use Lagrangian change of variables in order to handle the transport equation and to reduce the problem in the initial domain. However, the specificity here is that the Lagrangian coordinates do not coincide with the Eulerian coordinates at the boundaries since we consider slip boundary conditions. Therefore, it brings some extra nonlinear terms in the boundary conditions. The strategy is based on the study of the linearized system with nonhomogeneous boundary conditions and on the Banach fixed point theorem.The forward self-similar solution of fractional incompressible Navier-Stokes system: the critical casehttps://zbmath.org/1544.351162024-11-01T15:51:55.949586Z"Lai, Baishun"https://zbmath.org/authors/?q=ai:lai.baishunSummary: In this paper, we study the regularity and pointwise estimates of forward self-similar solutions of fractional Navier-Stokes system under the critical case. By employing a Caffarelli, Kohn and Nirenberg-type iteration, \(L^{\infty}\) estimates of the self-similar solution's profile are established, which is a key ingredient to ensure that the global weighted energy estimate procedure used in [\textit{B. Lai} et al., Trans. Am. Math. Soc. 374, No. 10, 7449--7497 (2021; Zbl 1479.35622)] is performed under the critical case. As a product, its natural pointwise bounds are recovered. Moreover, to obtain the optimal spatial decay estimate of self-similar solution's profile, a new technique is required due to lack of the related regularity theory.Local and global solvability for the Boussinesq system in Besov spaceshttps://zbmath.org/1544.351172024-11-01T15:51:55.949586Z"Yan, Shuokai"https://zbmath.org/authors/?q=ai:yan.shuokai"Wang, Lu"https://zbmath.org/authors/?q=ai:wang.lu.5|wang.lu.7|wang.lu|wang.lu.3|wang.lu.1|wang.lu.10"Zhang, Qinghua"https://zbmath.org/authors/?q=ai:zhang.qinghua(no abstract)On blowups of vorticity for the homogeneous Euler equationhttps://zbmath.org/1544.351182024-11-01T15:51:55.949586Z"Konopelchenko, B. G."https://zbmath.org/authors/?q=ai:konopelchenko.boris-g"Ortenzi, G."https://zbmath.org/authors/?q=ai:ortenzi.giovanniIn this paper, the authors consider the blow-up properties of vorticity for the three-and two-dimensional homogeneous Euler equations. Two regimes of approaching a blow-up point, respectively, with variable or fixed time are analyzed. It is shown that in the \(n\)-dimensional \((n=2, 3)\) generic case the blow-ups of degrees \(1,\dots,n\) at the variable time regime and of degrees \(1/2,\dots,(n+1)/(n+2)\) at the fixed time regime may exist. Particular situations when the vorticity blows while the direction of the vorticity vector is concentrated in one or two directions are realizable.
Reviewer: Mingqi Xiang (Tianjin)Boundary value and control problems for the stationary heat transfer model with variable coefficientshttps://zbmath.org/1544.351192024-11-01T15:51:55.949586Z"Baranovskii, Evgenii S."https://zbmath.org/authors/?q=ai:baranovskii.evgenii-sergeevich"Brizitskii, Roman V."https://zbmath.org/authors/?q=ai:brizitskii.roman-viktorovich|brizitskii.roman-victorovich"Saritskaia, Zhanna Yu."https://zbmath.org/authors/?q=ai:saritskaia.zhanna-yurievnaSummary: A stationary heat transfer model generalizing the Boussinesq approximation is considered. For the corresponding boundary value problem the property of a global existence of its weak solution is proved. We establish special requirements on the model data under which the maximum principle for temperature is valid. Moreover, the solvability property of the control problem for the examined heat transfer model is investigated.Stability analysis of gravity modulated thermosolutal convection in Casson fluid with internal heat sourcehttps://zbmath.org/1544.351202024-11-01T15:51:55.949586Z"Chandan, K. G."https://zbmath.org/authors/?q=ai:chandan.k-g"Akhila, P. A."https://zbmath.org/authors/?q=ai:akhila.p-a"Patil, Mallikarjun B."https://zbmath.org/authors/?q=ai:patil.mallikarjun-bSummary: The current work establishes research into the non-Newtonian Casson fluid's linear and weakly nonlinear stability analysis under the influences of internal heating and gravitational modulation. The study is confined to stationary convection, where marginal stability is determined by the critical Rayleigh number, which is derived from linear stability analysis. The marginal stability curves are plotted to observe the onset of convection due to different parameters that exist in the problem. Heat and mass transfer are measured in terms of the Nusselt number (Nu) and Sherwood number (Sh), respectively, in the nonlinear stability analysis. These measurements are based on the Ginzburg-Landau (GL) equation. One of the main outcomes of this stability analysis is that the internal Rayleigh number and Casson parameter behave similarly for mass transport in a fluid whereas it operates oppositely for heat transfer in a fluid.Hard congestion limit of the dissipative Aw-Rascle systemhttps://zbmath.org/1544.351212024-11-01T15:51:55.949586Z"Chaudhuri, N."https://zbmath.org/authors/?q=ai:chaudhuri.nilasis"Navoret, L."https://zbmath.org/authors/?q=ai:navoret.laurent"Perrin, C."https://zbmath.org/authors/?q=ai:perrin.charlotte"Zatorska, E."https://zbmath.org/authors/?q=ai:zatorska.ewelinaThe authors investigate the singular limit as \(\varepsilon \to 0\) for the following generalization of the Aw and Rascle traffic model
\[
\partial_t\rho_{\varepsilon} +\partial_x(\rho_{\varepsilon}u_{\varepsilon})=0; \ \partial_t(\rho_{\varepsilon}w_{\varepsilon})+\partial_x(\rho_{\varepsilon}u_{\varepsilon}w_{\varepsilon})=0,
\]
where \(\rho_{\varepsilon}\) represents the density and \(u_{\varepsilon}\) is the velocity of motion. The quantity \(w_{\varepsilon}\) denotes the desired velocity of motion and it differs from the actual velocity \(u_{\varepsilon}\) by the offset function, which describes the cost of moving and depends on the congestion of the flow. More precisely, we have
\[
w_{\varepsilon}=u_{\varepsilon} +\partial_xp_{\varepsilon}(\rho_{\varepsilon}),
\]
where
\[
p_{\varepsilon}(\rho_{\varepsilon})=\varepsilon \frac{F(\rho_{\varepsilon})}{(1-\rho_{\varepsilon})^{\beta}}, \mbox{with } F(s)\to 0 \text{ as } s\to 0, \ \beta >1.
\]
This singular function plays the role of a barrier by preventing the density to exceed the maximal threshold \(\bar \rho =1\). After proving the global existence of smooth solutions, the authors show the convergence of a subsequence of solutions towards a weak solution of a hybrid free-congested system. This is illustrated by numerical results.
Reviewer: Gheorghe Moroşanu (Cluj-Napoca)The compressible Navier-Stokes-Cahn-Hilliard equations with dynamic boundary conditionshttps://zbmath.org/1544.351222024-11-01T15:51:55.949586Z"Cherfils, Laurence"https://zbmath.org/authors/?q=ai:cherfils.laurence"Feireisl, Eduard"https://zbmath.org/authors/?q=ai:feireisl.eduard"Michálek, Martin"https://zbmath.org/authors/?q=ai:michalek.martin"Miranville, Alain"https://zbmath.org/authors/?q=ai:miranville.alain-m"Petcu, Madalina"https://zbmath.org/authors/?q=ai:petcu.madalina"Pražák, Dalibor"https://zbmath.org/authors/?q=ai:prazak.daliborSummary: We consider the compressible Navier-Stokes-Cahn-Hilliard system describing the behavior of a binary mixture of compressible, viscous and macroscopically immiscible fluids. The equations are endowed with dynamic boundary conditions which allows taking into account the interaction between the fluid components and the rigid walls of the physical domain. We establish the existence of global-in-time weak solutions for any finite energy initial data.Global gradient estimates for shear thinning-type Stokes system on non-smooth domainshttps://zbmath.org/1544.351232024-11-01T15:51:55.949586Z"Cho, Namkyeong"https://zbmath.org/authors/?q=ai:cho.namkyeongSummary: This article presents global \(L^q\) estimates for the weak solution of the steady \(p\)-Stokes equations, which describe the motion of shear-thinning flow under the nonslip boundary condition. We focus on non-smooth domains whose boundaries extend beyond the Lipschitz category, with coefficients belonging to the BMO (Bounded Mean Oscillation) space having a sufficiently small BMO semi-norm.Global well-posedness and exponential decay of strong solution to the three-dimensional nonhomogeneous Bénard system with density-dependent viscosity and vacuumhttps://zbmath.org/1544.351242024-11-01T15:51:55.949586Z"Li, Huanyuan"https://zbmath.org/authors/?q=ai:li.huanyuan"Liu, Jieqiong"https://zbmath.org/authors/?q=ai:liu.jieqiongSummary: In this paper, we are concerned with the three-dimensional nonhomogeneous Bénard system with density-dependent viscosity in bounded domain. The global well-posedness of strong solution is established, provided that the initial total mass \(\|\rho_0 \|_{L^1}\) is suitably small. In particular, the initial velocity and temperature can be arbitrarily large. Moreover, the exponential decay of strong solution is also obtained. It is worth noting that the vacuum of initial density is allowed.Global well-posedness of the 3D damped micropolar Bénard system with horizontal dissipationhttps://zbmath.org/1544.351252024-11-01T15:51:55.949586Z"Liu, Hui"https://zbmath.org/authors/?q=ai:liu.hui.5"Lin, Lin"https://zbmath.org/authors/?q=ai:lin.lin.2"Su, Dong"https://zbmath.org/authors/?q=ai:su.dong"Zhang, Qiangheng"https://zbmath.org/authors/?q=ai:zhang.qianghengSummary: In this paper, we consider the global well-posedness of the 3D damped micropolar Bénard system with horizontal dissipation. Global existence and uniqueness of the solution of system (1.1) are proved for \(\beta \geq 4\) and \(\alpha > 0\).Partial regularity and nonlinear potential estimates for Stokes systems with super-quadratic growthhttps://zbmath.org/1544.351262024-11-01T15:51:55.949586Z"Ma, Lingwei"https://zbmath.org/authors/?q=ai:ma.lingwei"Zhang, Zhenqiu"https://zbmath.org/authors/?q=ai:zhang.zhenqiuSummary: This paper builds a bridge between partial regularity theory and nonlinear potential theory for the following generalized stationary Stokes system with super-quadratic growth and continuous coefficients:
\[
-\operatorname{div} \mathcal{A}(x, D \boldsymbol{u}) + \nabla \pi = \boldsymbol{f} \quad \operatorname{div} \boldsymbol{u} = 0,
\]
where \(D\boldsymbol{u}\) is the symmetric part of the gradient \(\nabla \boldsymbol{u}\). We first establish an \(\varepsilon\)-regularity criterion involving both the excess functional of the symmetric gradient \(D\boldsymbol{u}\) and Wolff potentials of the nonhomogeneous term \(\boldsymbol{f}\) to guarantee the local vanishing mean oscillation (VMO)-regularity of \(D\boldsymbol{u}\) in an open subset \(\Omega_{\boldsymbol{u}}\) of \(\Omega\) with full measure. Such an \(\varepsilon\)-regularity criterion leads to a pointwise Wolff potential estimate of \(D\boldsymbol{u}\), which immediately infers that \(D\boldsymbol{u}\) is partially \(C^0\)-regular under appropriate assumptions. Finally, we give a local continuous modulus estimate of \(D\boldsymbol{u}\).On Onsager's type conjecture for the inviscid Boussinesq equationshttps://zbmath.org/1544.351272024-11-01T15:51:55.949586Z"Miao, Changxing"https://zbmath.org/authors/?q=ai:miao.changxing"Nie, Yao"https://zbmath.org/authors/?q=ai:nie.yao"Ye, Weikui"https://zbmath.org/authors/?q=ai:ye.weikuiSummary: In this paper, we investigate the Cauchy problem for the three dimensional inviscid Boussinesq system in the periodic setting. For \(1 \leq p \leq \infty\), we show that the threshold regularity exponent for \(L^{p}\)-norm conservation of temperature of this system is \(1/3\), consistent with Onsager exponent. More precisely, for \(1 \leq p \leq \infty\), every weak solution \((v, \theta) \in C_{t} C_{x}^{\beta}\) to the inviscid Boussinesq equations satisfies that \(\| \theta (t) \|_{L^{p} (\mathbb{T}^{3})} = \| \theta_{0} \|_{L^{p} (\mathbb{T}^{3})}\) if \(\beta > \frac{1}{3}\), while if \(\beta < \frac{1}{3}\), there exist infinitely many weak solutions \((v, \theta) \in C_{t} C_{x}^{\beta}\) such that the \(L^{p}\)-norm of temperature is not conserved. As a byproduct, we are able to construct many weak solutions in \(C_{t} C_{x}^{\beta}\) for \(\beta < \frac{1}{3}\) displaying wild behavior, such as fast kinetic energy dissipation and high oscillation of velocity. Moreover, we also show that if a weak solution \((v, \theta)\) of this system has at least one interval of regularity, then this weak solution \((v, \theta)\) is not unique in \(C_{t} C_{x}^{\beta}\) for \(\beta < \frac{1}{3}\).Topological defects on solutions of the non-relativistic equation for extended double ring-shaped potentialhttps://zbmath.org/1544.351282024-11-01T15:51:55.949586Z"Boudjedaa, Badredine"https://zbmath.org/authors/?q=ai:boudjedaa.badredine"Ahmed, Faizuddin"https://zbmath.org/authors/?q=ai:ahmed.faizuddin(no abstract)Conserved vectors and symmetry solutions of the Landau-Ginzburg-Higgs equation of theoretical physicshttps://zbmath.org/1544.351292024-11-01T15:51:55.949586Z"Khalique, Chaudry Masood"https://zbmath.org/authors/?q=ai:khalique.chaudry-masood"Lephoko, Mduduzi Yolane Thabo"https://zbmath.org/authors/?q=ai:lephoko.mduduzi-yolane-thabo(no abstract)Conformal structure of quantum wave mechanicshttps://zbmath.org/1544.351302024-11-01T15:51:55.949586Z"Petti, Richard James"https://zbmath.org/authors/?q=ai:petti.richard-jamesSummary: This work interprets the quantum terms in a Lagrangian, and consequently of the wave equation and momentum tensor, in terms of a modified spacetime metric. Part I interprets the quantum terms in the Lagrangian of a Klein-Gordon field as scalar curvature of conformal dilation covector nm that is proportional to \(\hbar\) times the gradient of wave amplitude \(R\). Part II replaces conformal dilation with a conformal factor \(\rho\) that defines a modified spacetime metric \(g^\prime = \exp(\rho)g\), where \(g\) is the gravitational metric. Quantum terms appear only in metric \(g^\prime\) and its metric connection coefficients. Metric \(g^\prime\) preserves lengths and angles in classical physics and in the domain of the quantum field itself. \(g^\prime\) combines concepts of quantum theory and spacetime geometry in one structure. The conformal factor can be interpreted as the limit of a distribution of inclusions and voids in a lattice that cause the metric to bulge or contract. All components of all free quantum fields satisfy the Klein-Gordon equation, so this interpretation extends to all quantum fields. Measurement operations, and elements of quantum field theory are not considered.Six types of spin solitons in three-component Bose-Einstein condensateshttps://zbmath.org/1544.351312024-11-01T15:51:55.949586Z"Wang, Yu-Hao"https://zbmath.org/authors/?q=ai:wang.yuhao"Meng, Ling-Zheng"https://zbmath.org/authors/?q=ai:meng.ling-zheng"Zhao, Li-Chen"https://zbmath.org/authors/?q=ai:zhao.li-chen(no abstract)Construction of conservation laws for the Gardner equation, Landau-Ginzburg-Higgs equation, and Hirota-Satsuma equationhttps://zbmath.org/1544.351322024-11-01T15:51:55.949586Z"Chen, Cheng"https://zbmath.org/authors/?q=ai:chen.cheng"Afzal, Faiza"https://zbmath.org/authors/?q=ai:afzal.faiza"Zhang, Yufeng"https://zbmath.org/authors/?q=ai:zhang.yu-feng(no abstract)Higher-dimensional integrable deformations of the classical Boussinesq-Burgers systemhttps://zbmath.org/1544.351332024-11-01T15:51:55.949586Z"Cheng, Xiaoyu"https://zbmath.org/authors/?q=ai:cheng.xiaoyu"Huang, Qing"https://zbmath.org/authors/?q=ai:huang.qing(no abstract)Decomposition solutions and Bäcklund transformations of the B-type and C-type Kadomtsev-Petviashvili equationshttps://zbmath.org/1544.351342024-11-01T15:51:55.949586Z"Hao, Xiazhi"https://zbmath.org/authors/?q=ai:hao.xiazhi"Lou, S. Y."https://zbmath.org/authors/?q=ai:lou.senyue(no abstract)Hybrid rogue waves and breather solutions on the double-periodic background for the Kundu-DNLS equationhttps://zbmath.org/1544.351352024-11-01T15:51:55.949586Z"Jiang, DongZhu"https://zbmath.org/authors/?q=ai:jiang.dongzhu"Zhaqilao"https://zbmath.org/authors/?q=ai:zhaqilao.z(no abstract)Nonlinear non-autonomous Boussinesq equationshttps://zbmath.org/1544.351362024-11-01T15:51:55.949586Z"Ludu, Andrei"https://zbmath.org/authors/?q=ai:ludu.andrei"Khanal, Harihar"https://zbmath.org/authors/?q=ai:khanal.harihar"Carstea, Adrian Stefan"https://zbmath.org/authors/?q=ai:carstea.adrian-stefanSummary: We study solitary wave solutions for a nonlinear and non-autonomous Boussinesq system with initial conditions. Since the variable coefficients introduce distortions and modulations of the solution amplitudes, we implement a multiple-scale approach combining various modes in order to capture the coupling between the nonlinear evolution and the effect of the variable coefficient. The differential system is mapped into a solvable system of nonlinear and non-autonomous ODE which is integrable by recursion procedures. We show that even in the limiting autonomous case, the multiple-scale approach gives a new possibly integrable dispersionless coupled envelope system, which deserves further study. We validate our theoretical results with numerical simulations, and we study their stability.Spatial structure of the non-integrable discrete defocusing Hirota equationhttps://zbmath.org/1544.351372024-11-01T15:51:55.949586Z"Ma, Liyuan"https://zbmath.org/authors/?q=ai:ma.liyuan"Fang, Miaoshuang"https://zbmath.org/authors/?q=ai:fang.miaoshuang"Song, Haifang"https://zbmath.org/authors/?q=ai:song.haifang"Zhou, Jiali"https://zbmath.org/authors/?q=ai:zhou.jiali(no abstract)New interaction solutions of the (2+1)-dimensional Nizhnik-Novikov-Veselov-type system and fusion phenomenahttps://zbmath.org/1544.351382024-11-01T15:51:55.949586Z"Wang, Guo-Hua"https://zbmath.org/authors/?q=ai:wang.guohua"Lin, Ji"https://zbmath.org/authors/?q=ai:lin.ji.2|lin.ji"Shen, Shou-Feng"https://zbmath.org/authors/?q=ai:shen.shoufeng(no abstract)New patterns of localized excitations in (2+1)-dimensions: the fifth-order asymmetric Nizhnik-Novikov-Veselov equationhttps://zbmath.org/1544.351392024-11-01T15:51:55.949586Z"Wang, Jianyong"https://zbmath.org/authors/?q=ai:wang.jianyong"Chai, Yuanhua"https://zbmath.org/authors/?q=ai:chai.yuanhua(no abstract)Rogue waves for the (2+1)-dimensional Myrzakulov-Lakshmanan-IV equation on a periodic backgroundhttps://zbmath.org/1544.351402024-11-01T15:51:55.949586Z"Wang, Xiao-Hui"https://zbmath.org/authors/?q=ai:wang.xiaohui"Zhaqilao"https://zbmath.org/authors/?q=ai:zhaqilao.z(no abstract)On the analytical soliton approximations to fractional forced Korteweg-de Vries equation arising in fluids and plasmas using two novel techniqueshttps://zbmath.org/1544.351412024-11-01T15:51:55.949586Z"Alhejaili, Weaam"https://zbmath.org/authors/?q=ai:alhejaili.weaam"Az-Zo'bi, Emad A."https://zbmath.org/authors/?q=ai:az-zobi.emad-a"Shah, Rasool"https://zbmath.org/authors/?q=ai:shah.rasool"El-Tantawy, S. A."https://zbmath.org/authors/?q=ai:el-tantawy.s-a(no abstract)The Mumford dynamical system and the Gelfand-Dikii recursionhttps://zbmath.org/1544.351422024-11-01T15:51:55.949586Z"Baron, P. G."https://zbmath.org/authors/?q=ai:baron.p-gIt is difficult to present the contents of this paper, so I simply reproduce below the abstract of the paper which might be useful for interested people.
In his paper [Funct. Anal. Appl. 57, No. 4, 288--302 (2023; Zbl 1540.35336); translation from Funkts. Anal. Prilozh. 57, No. 4, 27--45 (2023)], \textit{V. M. Buchstaber} developed the differential-algebraic theory of the Mumford dynamical system. The key object of this theory is the (P, Q)-recursion introduced in his paper. In the present paper, we further develop the theory of the (P, Q)-recursion and describe its connections to the Korteweg-de Vries hierarchy, the Lenard operator, and the Gelfand-Dikii recursion.
Reviewer: Gheorghe Moroşanu (Cluj-Napoca)An integral equation formalism for integrating a nonlinear initial-boundary value problem for a Boussinesq equationhttps://zbmath.org/1544.351432024-11-01T15:51:55.949586Z"Jang, T. S."https://zbmath.org/authors/?q=ai:jang.taek-soo|jang.tae-seokSummary: In this paper, a new nonlinear initial-\textit{boundary} value problem for a Boussinesq equation is formulated. And a coupled system of nonlinear integral equations, equivalent to the new initial-boundary value problem, is constructed for integrating the initial-\textit{boundary} value problem, but which is \textit{inherently} different from other conventional formulations for integral equations. For the numerical solutions, successive approximations are applied, which leads to a functional iterative formula. A propagating solitary wave is simulated via iterating the formula, which is in good agreement with the known exact solution.Breather wave solutions on the Weierstrass elliptic periodic background for the \((2 + 1)\)-dimensional generalized variable-coefficient KdV equationhttps://zbmath.org/1544.351442024-11-01T15:51:55.949586Z"Li, Jiabin"https://zbmath.org/authors/?q=ai:li.jiabin"Yang, Yunqing"https://zbmath.org/authors/?q=ai:yang.yunqing"Sun, Wanyi"https://zbmath.org/authors/?q=ai:sun.wanyi(no abstract)Ren-integrable and Ren-symmetric integrable systemshttps://zbmath.org/1544.351452024-11-01T15:51:55.949586Z"Lou, S. Y."https://zbmath.org/authors/?q=ai:lou.senyue(no abstract)Exact solitary wave solutions of the (3+1)-dimensional generalised Kadomtsev-Petviashvili Benjamin-Bona-Mahony equationhttps://zbmath.org/1544.351462024-11-01T15:51:55.949586Z"Mallick, Biswajit"https://zbmath.org/authors/?q=ai:mallick.biswajit"Sahu, Prakash Kumar"https://zbmath.org/authors/?q=ai:sahu.prakash-kumarSummary: In this paper, the authors studied the (3+1)-dimensional generalised Kadomtsev-Petviashvili Benjamin-Bona-Mahony (KP-BBM) equation. The exact solitary wave solutions of the nonlinear partial differential equation have been obtained by using the modified hyperbolic function expansion method. The required preliminaries for the method of modified hyperbolic expansion have been given. Two numerical examples have been illustrated and the exact solutions obtained are explained with the help of two-dimensional and three-dimensional graphs. The solutions have different physical structures depending on the involved parameters. This technique can also be employed to compute the new exact solution of higher order nonlinear partial differential equations.
For the entire collection see [Zbl 1522.00198].Nonlocal symmetries, soliton-cnoidal wave solution and soliton molecules to a (2+1)-dimensional modified KdV systemhttps://zbmath.org/1544.351472024-11-01T15:51:55.949586Z"Wang, Jianyong"https://zbmath.org/authors/?q=ai:wang.jianyong"Ren, Bo"https://zbmath.org/authors/?q=ai:ren.bo(no abstract)Traveling wave solution of the Olver-Rosenau equation solved by dynamics systemhttps://zbmath.org/1544.351482024-11-01T15:51:55.949586Z"Xiong, Mei"https://zbmath.org/authors/?q=ai:xiong.mei"Chen, Longwei"https://zbmath.org/authors/?q=ai:chen.longwei"Yang, Na"https://zbmath.org/authors/?q=ai:yang.naSummary: Olver-Rosenau equations presented by Olver and Rosenau can be rewritten to the dynamic system by the wave transformation. The system is a Hamiltonian system with the first integral, and its phase-space and equilibrium point analysis are given in different parameter spaces in detail. On this basis, we can derive various solutions of the original equation relating these orbits in different phase-space planes, and the theoretical basis of the numerical solution is provided for engineering application and production practice.Scattering threshold for the focusing energy-critical generalized Hartree equationhttps://zbmath.org/1544.351492024-11-01T15:51:55.949586Z"Almuthaybiri, Saleh"https://zbmath.org/authors/?q=ai:almuthaybiri.saleh-salhan-g"Peng, Congming"https://zbmath.org/authors/?q=ai:peng.congming"Saanouni, Tarek"https://zbmath.org/authors/?q=ai:saanouni.tarek(no abstract)Mass-energy threshold dynamics for the focusing NLS with a repulsive inverse-power potentialhttps://zbmath.org/1544.351502024-11-01T15:51:55.949586Z"Ardila, Alex H."https://zbmath.org/authors/?q=ai:ardila.alex-hernandez"Hamano, Masaru"https://zbmath.org/authors/?q=ai:hamano.masaru"Ikeda, Masahiro"https://zbmath.org/authors/?q=ai:ikeda.masahiroSummary: In this paper we study long time dynamics (i.e., scattering and blow-up) of solutions for the focusing NLS with a repulsive inverse-power potential and with initial data lying exactly at the mass-energy threshold, namely, when \(E_V (u_0) M(u_0) = E_0 (Q)M(Q)\). Moreover, we prove failure of the uniform space-time bounds at the mass-energy threshold.Stability analysis and soliton solutions of the (1+1)-dimensional nonlinear chiral Schrödinger equation in nuclear physicshttps://zbmath.org/1544.351512024-11-01T15:51:55.949586Z"Badshah, Fazal"https://zbmath.org/authors/?q=ai:badshah.fazal"Tariq, Kalim U."https://zbmath.org/authors/?q=ai:tariq.kalim-u"Bekir, Ahmet"https://zbmath.org/authors/?q=ai:bekir.ahmet"Kazmi, S. M. Raza"https://zbmath.org/authors/?q=ai:kazmi.s-m-raza"Az-Zo'bi, Emad"https://zbmath.org/authors/?q=ai:az-zobi.emad-a(no abstract)Generalized and multi-oscillation solitons in the nonlinear Schrödinger equation with quartic dispersionhttps://zbmath.org/1544.351522024-11-01T15:51:55.949586Z"Bandara, Ravindra"https://zbmath.org/authors/?q=ai:bandara.ravindra"Giraldo, Andrus"https://zbmath.org/authors/?q=ai:giraldo.andrus"Broderick, Neil G. R."https://zbmath.org/authors/?q=ai:broderick.neil-g-r"Krauskopf, Bernd"https://zbmath.org/authors/?q=ai:krauskopf.bernd(no abstract)Focusing nonlocal nonlinear Schrödinger equation with asymmetric boundary conditions: large-time behaviorhttps://zbmath.org/1544.351532024-11-01T15:51:55.949586Z"Boutet de Monvel, Anne"https://zbmath.org/authors/?q=ai:boutet-de-monvel.anne-marie"Rybalko, Yan"https://zbmath.org/authors/?q=ai:rybalko.yan"Shepelsky, Dmitry"https://zbmath.org/authors/?q=ai:shepelskyi.dmytro-georgiiovychThe purpose of this paper is to study the initial value problem for the focusing nonlocal nonlinear Schrödinger equation
\[
\begin{cases}
iu_{t}\left( x,t\right) +u_{xx}\left( x,t\right) +2u^{2}\left(x,t\right) \overline{u\left( -x,t\right) }=0,\ x\in\mathbb{R}\ t\in\mathbb{R},\\
u\left( x,0\right) =u_{0}\left( x\right),\ x\in\mathbb{R},
\end{cases} \tag{1}
\]
(\(\overline{u}\) denotes the complex conjugate of \(u\)) with asymmetric nonzero boundary conditions:
\[
u\left( x,t\right) \rightarrow\pm Ae^{-2iA^{2}t},\quad x\rightarrow\pm\infty,\ t\in\mathbb{R}.
\]
It is shown that for a class of initial data there exist three qualitatively different asymptotic zones in the \(\left( x,t\right)\) plane: the regions \(\left\vert \frac{x}{4t}\right\vert \in\left( A/2,\infty\right),\) where the parameters are modulated, i.e., they depend on the ratio \(\frac{x}{t},\) and a central region \(\left\vert \frac{x}{4t}\right\vert \in\left( 0,A/2\right)\), where the parameters are unmodulated. The asymptotics of the solution as \(\xi\rightarrow\pm0\), for fixed \(x=x_{0}>0\) and \(t\rightarrow\infty\) is also considered. The method used in this paper is based on the inverse scattering transform, that allows to express the solution of problem (1) in terms of the solution of an associated Riemann-Hilbert problem, and the Deift and Zhou nonlinear steepest descent method in [\textit{P. A. Deift} et al., in: Important developments in soliton theory. Berlin: Springer-Verlag. 181--204 (1993; Zbl 0926.35132); \textit{P. Deift} and \textit{X. Zhou}, Ann. Math. (2) 137, No. 2, 295--368 (1993; Zbl 0771.35042)].
For the entire collection see [Zbl 1519.47002].
Reviewer: Ivan Naumkin (Ciudad de México)Rogue wave solutions and rogue-breather solutions to the focusing nonlinear Schrödinger equationhttps://zbmath.org/1544.351542024-11-01T15:51:55.949586Z"Chen, Si-Jia"https://zbmath.org/authors/?q=ai:chen.sijia"Lü, Xing"https://zbmath.org/authors/?q=ai:lu.xing(no abstract)A general coupled derivative nonlinear Schrödinger system: Darboux transformation and soliton solutionshttps://zbmath.org/1544.351552024-11-01T15:51:55.949586Z"Kuang, Yonghui"https://zbmath.org/authors/?q=ai:kuang.yonghuiSummary: In this work we present a general coupled derivative nonlinear Schrödinger system. We construct the corresponding \(N\)-fold Darboux transform and generalized Darboux transform. Under this construction, we give different soliton solutions and plot their figures describing the soliton characteristics and dynamical behaviors, including higher-order soliton and rouge wave solution etc.Deformation of optical solitons in a variable-coefficient nonlinear Schrödinger equation with three distinct \(\mathcal{PT}\)-symmetric potentials and modulated nonlinearitieshttps://zbmath.org/1544.351562024-11-01T15:51:55.949586Z"Manikandan, K."https://zbmath.org/authors/?q=ai:manikandan.kannan"Sakkaravarthi, K."https://zbmath.org/authors/?q=ai:sakkaravarthi.k"Sudharsan, J. B."https://zbmath.org/authors/?q=ai:sudharsan.j-b"Aravinthan, D."https://zbmath.org/authors/?q=ai:aravinthan.d(no abstract)Binary Darboux transformation of vector nonlocal reverse-space nonlinear Schrödinger equationshttps://zbmath.org/1544.351572024-11-01T15:51:55.949586Z"Ma, Wen-Xiu"https://zbmath.org/authors/?q=ai:ma.wen-xiu"Huang, Yehui"https://zbmath.org/authors/?q=ai:huang.yehui"Wang, Fudong"https://zbmath.org/authors/?q=ai:wang.fudong"Zhang, Yong"https://zbmath.org/authors/?q=ai:zhang.yong.18|zhang.yong|zhang.yong.54|zhang.yong.2|zhang.yong.13|zhang.yong.12|zhang.yong.9|zhang.yong.41|zhang.yong.62|zhang.yong.67|zhang.yong.10|zhang.yong.8|zhang.yong.28|zhang.yong.59|zhang.yong.15|zhang.yong.5|zhang.yong.52|zhang.yong.57|zhang.yong.60|zhang.yong.64|zhang.yong.14|zhang.yong.19|zhang.yong.4"Ding, Liyuan"https://zbmath.org/authors/?q=ai:ding.liyuanSummary: For vector nonlocal reverse-space nonlinear Schrödinger equations, a binary Darboux transformation is formulated by using two sets of eigenfunctions and adjoint eigenfunctions. The resulting binary Darboux transformation has been decomposed into an \(N\)-fold product of single binary Darboux transformations. An application starting from zero seed potentials generates a class of soliton solutions.Corrigendum to: ``Global well-posedness and scattering of the defocusing energy-critical inhomogeneous nonlinear Schrödinger equation with radial data''https://zbmath.org/1544.351582024-11-01T15:51:55.949586Z"Park, Dongjin"https://zbmath.org/authors/?q=ai:park.dongjinCorrigendum to the author's paper [ibid. 536, No. 2, Article ID 128202, 28 p. (2024; Zbl 1536.35309)].Stability of black solitons in optical systems with intensity-dependent dispersionhttps://zbmath.org/1544.351592024-11-01T15:51:55.949586Z"Pelinovsky, Dmitry E."https://zbmath.org/authors/?q=ai:pelinovsky.dmitry-e"Plum, Michael"https://zbmath.org/authors/?q=ai:plum.michaelAuthors' abstract: Black solitons are identical in the nonlinear Schrödinger (NLS) equation with intensity-dependent dispersion and the cubic defocusing NLS equation. We prove that the intensity-dependent dispersion introduces new properties in the stability analysis of the black soliton. First, the spectral stability problem possesses only isolated eigenvalues on the imaginary axis. Second, the energetic stability argument holds in Sobolev spaces with exponential weights. Third, the black soliton persists with respect to the addition of a small decaying potential and remains spectrally stable when it is pinned to the minimum points of the effective potential. The same model exhibits a family of traveling dark solitons for every wave speed and we incorporate properties of these dark solitons for small wave speeds in the analysis of orbital stability of the black soliton.
Reviewer: Alessandro Selvitella (Fort Wayne)The quasi-Gramian solution of a non-commutative extension of the higher-order nonlinear Schrödinger equationhttps://zbmath.org/1544.351602024-11-01T15:51:55.949586Z"Riaz, H. W. A."https://zbmath.org/authors/?q=ai:riaz.h-wajahat-ahmed.2"Lin, J."https://zbmath.org/authors/?q=ai:lin.ji.1(no abstract)Transverse instability in nonparaxial systems with four-wave mixinghttps://zbmath.org/1544.351612024-11-01T15:51:55.949586Z"Tamilselvan, K."https://zbmath.org/authors/?q=ai:tamilselvan.k"Govindarajan, A."https://zbmath.org/authors/?q=ai:govindarajan.aravind"Senthil Pandian, M."https://zbmath.org/authors/?q=ai:senthil-pandian.m"Ramasamy, P."https://zbmath.org/authors/?q=ai:ramasamy.p(no abstract)Cauchy matrix approach to three non-isospectral nonlinear Schrödinger equationshttps://zbmath.org/1544.351622024-11-01T15:51:55.949586Z"Tefera, Alemu Yilma"https://zbmath.org/authors/?q=ai:tefera.alemu-yilma"Li, Shangshuai"https://zbmath.org/authors/?q=ai:li.shangshuai"Zhang, Da-jun"https://zbmath.org/authors/?q=ai:zhang.dajun(no abstract)Beyond-band discrete soliton interaction in binary waveguide arrayshttps://zbmath.org/1544.351632024-11-01T15:51:55.949586Z"Tran, Minh C."https://zbmath.org/authors/?q=ai:tran.minh-cong"Tran, Truong X."https://zbmath.org/authors/?q=ai:tran.truong-x(no abstract)The exact solutions for the non-isospectral Kaup-Newell hierarchy via the inverse scattering transformhttps://zbmath.org/1544.351642024-11-01T15:51:55.949586Z"Zhang, Hongyi"https://zbmath.org/authors/?q=ai:zhang.hongyi"Zhang, Yufeng"https://zbmath.org/authors/?q=ai:zhang.yu-feng"Feng, Binlu"https://zbmath.org/authors/?q=ai:feng.binlu"Afzal, Faiza"https://zbmath.org/authors/?q=ai:afzal.faizaSummary: We begin by introducing a non-isospectral Lax pair, from which we derive a non-isospectral integrable Kaup-Newell hierarchy. The new general solutions for the non-isospectral integrable Kaup-Newell hierarchy are obtained through the inverse scattering transform (IST) method. Finally, we obtain the soliton solutions of a reduced non-isospectral integrable equation from non-isospectral integrable Kaup-Newell hierarchy. For 1-soliton solution, we obtain the explicit expression and analyze the dynamical behavior of soliton solution; for 2-soliton solutions, we verify that collisions between soliton solutions are inelastic collisions. The significant difference of the paper from the works finished by other authors (such as
[\textit{T.-k. Ning} et al., Chaos Solitons Fractals 21, No. 2, 395--401 (2004; Zbl 1049.35160);
Physica A 339, 248--266 (2004; \url{doi:10.1016/j.physa.2004.03.021});
\textit{Q. Li} et al., J. Phys. A, Math. Theor. 41, No. 35, Article ID 355209, 14 p. (2008; Zbl 1158.35418);
Commun. Theor. Phys. 54, No. 2, 219--228 (2010; Zbl 1219.35240);
Chaos Solitons Fractals 45, No. 12, 1479--1485 (2012; Zbl 1258.37066)]) lies in except for the spectral parameter \(\lambda\) being higher order, not one-order of \(\lambda\).PT-symmetric solitons and parameter discovery in self-defocusing saturable nonlinear Schrödinger equation via LrD-PINNhttps://zbmath.org/1544.351652024-11-01T15:51:55.949586Z"Zhu, Bo-Wei"https://zbmath.org/authors/?q=ai:zhu.bo-wei"Bo, Wen-Bo"https://zbmath.org/authors/?q=ai:bo.wen-bo"Cao, Qi-Hao"https://zbmath.org/authors/?q=ai:cao.qi-hao"Geng, Kai-Li"https://zbmath.org/authors/?q=ai:geng.kai-li"Wang, Yue-Yue"https://zbmath.org/authors/?q=ai:wang.yueyue"Dai, Chao-Qing"https://zbmath.org/authors/?q=ai:dai.chaoqing(no abstract)Characterization of time-dependence for dissipative solitons stabilized by nonlinear gradient terms: periodic and quasiperiodic vs chaotic behaviorhttps://zbmath.org/1544.351662024-11-01T15:51:55.949586Z"Descalzi, Orazio"https://zbmath.org/authors/?q=ai:descalzi.orazio"Facão, M."https://zbmath.org/authors/?q=ai:facao.margarida"Cartes, Carlos"https://zbmath.org/authors/?q=ai:cartes.carlos"Carvalho, M. I."https://zbmath.org/authors/?q=ai:carvalho.m-i"Brand, Helmut R."https://zbmath.org/authors/?q=ai:brand.helmut-r(no abstract)On self-similar patterns in coupled parabolic systems as non-equilibrium steady stateshttps://zbmath.org/1544.351672024-11-01T15:51:55.949586Z"Mielke, Alexander"https://zbmath.org/authors/?q=ai:mielke.alexander"Schindler, Stefanie"https://zbmath.org/authors/?q=ai:schindler.stefanieThe authors investigate reaction-diffusion systems and other related dissipative systems on unbounded domains. They discuss the self-similarity and asymptotical behavior. Self-similar behavior is a known phenomenon in extended systems. The solutions are usually considered with trivial behavior at infinity in the case of finite mass or energy of the considered physical system.
The authors of the present paper discuss the asymptotic self-similar behavior and show that it can occur in three different ways. Given three cases which can be distinguished after transforming into scaling variables: (A) The transformed system is autonomous and its steady state is a classical self-similar solution. (B) The transformed system converges to an autonomous system having suitable steady states. (C) An exponentially growing term generates a constraint that generates a constrained steady state. Constrained self-similar profiles occur mainly in systems of PDEs where diffusion and reaction terms scale differently.
It seems that the authors suppose that case (A) often occurs in scalar equations while cases (B) and (C) are more common in coupled systems of equations. Moreover, they consider the case of nonzero boundary conditions at infinity, which leads to systems with infinite mass displaying a richer structure than finite-mass systems.
For the evolution equation \((1)\) \(\tilde{u}_t=\tilde{f}(t,x, \tilde{u},\tilde{\nabla}\tilde{u}. \ldots ,\tilde{\nabla}^k\tilde{u})\) per definition, the solution \(\tilde{u}\) is called self-similar, in the sense of Barenblatt, if it can be written in the form \(\tilde{u}(t,x)= (1+t)^{-\alpha}U(x/(1+t)^{\beta})\) for a function \(U\) and scaling exponents \(\alpha \) and \(\beta \), which are suitably chosen, for instance, in order to guarantee mass conservation. The function \(U\) is called the profile of the self-similar solution. This concept is well-known, as it already finds an application for simple problems. To classify different types of self-similarity one has to transform the system into suitable scaling coordinates in which the self-similar profile appears as a steady pattern. The new coordinates \((\tau ,y)\) can be obtained by \(\tau = \log{(1+t)}\) and \(y=x/(1+t)^{\beta}\). After replacing \(u(\tau ,y)= (1+t)^{\alpha}\tilde{u}(t,x)\) in \((1)\) then obtain \((2)\) \(u_{\tau}=f(\tau ,y,u,\nabla u, \ldots ,\nabla^{k}u)\). Here \(\nabla\) concerns spatial derivatives w.r.t. \(y\). When the general structure of the transformed system has the form \(\boldsymbol{u}_{\tau}=\boldsymbol{f} (y,\boldsymbol{u},\nabla\boldsymbol{u}, \ldots ,\nabla^k\boldsymbol{u})+ e^{\gamma\tau} \boldsymbol{g} (y,\boldsymbol{u},\nabla\boldsymbol{u}, \ldots ,\nabla^k\boldsymbol{u})\), \(\gamma > 0\), and if \(\boldsymbol{u}(\tau ,y)\to \boldsymbol{u}(y)\) as \(\tau\to\infty\), then the constrained self-similar profile \(\boldsymbol{u}\) should satisfy \(\boldsymbol{g} (y,\boldsymbol{u},\nabla\boldsymbol{u}, \ldots ,\nabla^k\boldsymbol{u})=0\) and \(\boldsymbol{f} (y,\boldsymbol{u},\nabla\boldsymbol{u}, \ldots ,\nabla^k\boldsymbol{u})+ \boldsymbol{\lambda }(y)=0\), \(y\in\mathbb{R}^d\).
Next the authors study systems on the unbounded real line that have the property that their restriction to a finite domain has a Lyapunov function and a gradient structure. Then the system reach local equilibrium on a rather fast time scale, but on unbounded domains with an infinite amount of mass or energy, it leads to a persistent mass or energy flow. It turns out that no true equilibrium is reached globally. In suitably rescaled variables, however, the solutions to the transformed system converge to non-equilibrium steady states that correspond to asymptotically self-similar behavior in the original system.
Reviewer: Dimitar A. Kolev (Sofia)Nonlinear dynamic wave characteristics of optical soliton solutions in ion-acoustic wavehttps://zbmath.org/1544.351682024-11-01T15:51:55.949586Z"Zaman, U. H. M."https://zbmath.org/authors/?q=ai:zaman.u-h-m"Arefin, Mohammad Asif"https://zbmath.org/authors/?q=ai:arefin.mohammad-asif"Hossain, Md. Akram"https://zbmath.org/authors/?q=ai:hossain.md-akram"Akbar, M. Ali"https://zbmath.org/authors/?q=ai:ali-akbar.m"Uddin, M. Hafiz"https://zbmath.org/authors/?q=ai:hafiz-uddin.mSummary: Traveling wave solutions are utilized to depict reaction-diffusion and analyze electrical signal transmission and propagation in space-time nonlinear fractional order partial differential equations like the space-time fractional Telegraph and Kolmogorov-Petrovsky Piskunov equations, and that are used in physical science to model combustion, biological research to model nerve impulse propagation, chemical dynamics to model concentration in order wave propagation, and plasma to model the progression of a set of duffing oscillators. In this study, the new generalized \((G^\prime / G)\)-expansion technique was employed to construct some novel and more universal closed-form traveling wave solutions in the sense of conformable derivatives which explain the above-stated phenomena properly. By utilizing complex fractional transformation, the ordinary differential equations are generated from fractional order differential equations. The recommended technique allowed us to produce some dynamical wave patterns of kink, single soliton, compacton, periodic shape, multiple periodic waves, anti-kink, and other structures are developed, which are shown using 3D plots and contour plots to illustrate the physical layout clearly. The traveling waveform responses can be defined in terms of functions based on trigonometry, hyperbolic operations, and rational functions and that are quick, flexible, and simple to reproduce. Furthermore, the obtained closed-form solutions for nonlinear fractional evolution equations make stability analysis and accuracy comparison amongst numerical solvers easier, which asserts that the new generalized \((G^\prime / G)\)-expansion technique is one of the most proficient and effective approaches.Optimal design of large-scale nonlinear Bayesian inverse problems under model uncertaintyhttps://zbmath.org/1544.351692024-11-01T15:51:55.949586Z"Alexanderian, Alen"https://zbmath.org/authors/?q=ai:alexanderian.alen"Nicholson, Ruanui"https://zbmath.org/authors/?q=ai:nicholson.ruanui"Petra, Noemi"https://zbmath.org/authors/?q=ai:petra.noemiSummary: We consider optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by partial differential equations (PDEs) under model uncertainty. Specifically, we consider inverse problems in which, in addition to the inversion parameters, the governing PDEs include secondary uncertain parameters. We focus on problems with infinite-dimensional inversion and secondary parameters and present a scalable computational framework for optimal design of such problems. The proposed approach enables Bayesian inversion and OED under uncertainty within a unified framework. We build on the Bayesian approximation error (BAE) approach, to incorporate modeling uncertainties in the Bayesian inverse problem, and methods for A-optimal design of infinite-dimensional Bayesian nonlinear inverse problems. Specifically, a Gaussian approximation to the posterior at the maximum \textit{a posteriori} probability point is used to define an uncertainty aware OED objective that is tractable to evaluate and optimize. In particular, the OED objective can be computed at a cost, in the number of PDE solves, that does not grow with the dimension of the discretized inversion and secondary parameters. The OED problem is formulated as a binary bilevel PDE constrained optimization problem and a greedy algorithm, which provides a pragmatic approach, is used to find optimal designs. We demonstrate the effectiveness of the proposed approach for a model inverse problem governed by an elliptic PDE on a three-dimensional domain. Our computational results also highlight the pitfalls of ignoring modeling uncertainties in the OED and/or inference stages.
{{\copyright} 2024 IOP Publishing Ltd}Some gradient theories in linear visco-elastodynamics towards dispersion and attenuation of waves in relation to large-strain modelshttps://zbmath.org/1544.351702024-11-01T15:51:55.949586Z"Roubíček, Tomáš"https://zbmath.org/authors/?q=ai:roubicek.tomasSummary: Various spatial-gradient extensions of standard viscoelastic rheologies of the Kelvin-Voigt, Maxwell's, and Jeffreys' types are analysed in linear one-dimensional situations as far as the propagation of waves and their dispersion and attenuation. These gradient extensions are then presented in the large-strain nonlinear variants where they are sometimes used rather for purely analytical reasons either in the Lagrangian or the Eulerian formulations without realizing this wave propagation context. The interconnection between these two modelling aspects is thus revealed in particular selected cases.Finite-strain poro-visco-elasticity with degenerate mobilityhttps://zbmath.org/1544.351712024-11-01T15:51:55.949586Z"van Oosterhout, Willem J. M."https://zbmath.org/authors/?q=ai:van-oosterhout.willem-j-m"Liero, Matthias"https://zbmath.org/authors/?q=ai:liero.matthiasThe authors consider a time horizon \(T>0\), a bounded and open domain \(\Omega \subseteq \mathbb{R}^{d}\), and the quasi-static system: \(-\operatorname{div}(\sigma _{el}(\nabla \chi ,c)+\sigma _{vi}(\nabla \chi ,\nabla \overset{.}{\chi } ,c)-\operatorname{div}\mathfrak{h}(D^{2}\chi ))=f(t)\), \(\overset{.}{c}-\operatorname{div}(\mathcal{M} (\nabla \chi ,c)\nabla \mu )=0\), in \([0,T]\times \Omega \), the total stress \( \Sigma _{tot}=\sigma _{el}+\sigma _{vi}-\operatorname{div}\mathfrak{h}\) consisting of the elastic stress \(\sigma _{el}(F,c)=\partial _{F}\Phi (F,c)\), the viscous stress \(\sigma _{vi}(F,\overset{.}{F},c)=\partial _{\overset{.}{F}}\zeta (F, \overset{.}{F},c)\), and the hyperstress \(\mathfrak{h}(G)=\partial _{G} \mathcal{H}(G)\), \(f\) is a body force, \(\mathcal{M}\) is the mobility tensor, and \(\mu (F,c)=\partial _{c}\Phi (F,c)\) is the chemical potential. The authors give two examples of such poro-visco-elastic materials. The boundary conditions \(\chi =Id\), on \(\Gamma _{D}\), \((\sigma _{el}(\nabla \chi ,c)+\sigma _{vi}(\nabla \chi ,\nabla \overset{.}{\chi },c)-\operatorname{div}_{x}(\mathfrak{ h}(D^{2}\chi ))\cdot \overrightarrow{n}=g(t)\), on \(\Gamma _{N}\), \(\mathfrak{h }(D^{2}\chi ):(\overrightarrow{n}\otimes \overrightarrow{n})=0\), on \( \partial \Omega \), and \(\mathcal{M}(\nabla \chi ,c)\nabla \mu \cdot \overrightarrow{n}=\kappa (\mu _{ext}(t)-\mu )\), on \(\partial \Omega \), are added, where \(\overrightarrow{n}\) denotes the unit normal vector on \( \partial \Omega \), \(\kappa \geq 0\) is a given permeability and \(\mu _{ext}\) is an external potential. The authors define a weak solution to this problem as a pair \((\chi ,c)\) with \(\chi \in L^{\infty }(0,T;W_{id}^{2,p}(\Omega ; \mathbb{R}^{d}))\), \(\overset{.}{\chi }\in L^{2}(0,T;H^{1}(\Omega ;\mathbb{R} ^{d}))\) and \(c\in L^{\infty }(0,T;L\log L(\Omega )))\), \(\overset{.}{c}\in L^{s}(0,T;W^{1,ss}(\Omega )^{\ast })\), and \(\nabla c^{m/2}\in L^{2}(0,T;L^{2}(\Omega ))\), and depending on the hypotheses additionally \( c\in L^{\infty }(0,T;L^{2+r}(\Omega ))\) and \(\nabla c^{m/2+1+r}\in L^{2}(0,T;L^{2}(\Omega ))\), the pair \((\chi ,c)\) satisfying variational formulations associated with the previous equations. The main result proves under hypotheses on the data the existence of a weak solution to this problem. For the proof, the authors introduce a regularized problem and a time discretization through a modified variational formulation. They prove the existence of a solution to this time-discretized and regularized problem, applying Schauder's fixed-point theorem. They then prove an energy-dissipation inequality and uniform estimates with respect to the regularizing and time discretization parameters, which allow passing to the limit.
Reviewer: Alain Brillard (Riedisheim)Stability of two weakly coupled elastic beams with partially local dampinghttps://zbmath.org/1544.351722024-11-01T15:51:55.949586Z"Zhang, Caihong"https://zbmath.org/authors/?q=ai:zhang.caihong"Huang, Yinuo"https://zbmath.org/authors/?q=ai:huang.yinuo"Wang, Licheng"https://zbmath.org/authors/?q=ai:wang.licheng"Duan, Chongxiong"https://zbmath.org/authors/?q=ai:duan.chongxiong"Zhang, Tiezhu"https://zbmath.org/authors/?q=ai:zhang.tiezhu"Wang, Kai"https://zbmath.org/authors/?q=ai:wang.kai.8|wang.kai.9|wang.kai.20|wang.kai.2|wang.kai.22|wang.kai.10|wang.kai.15|wang.kai.5|wang.kai.4|wang.kai.11|wang.kai.1|wang.kai.17|wang.kai.14|wang.kai.3|wang.kai.6Summary: In this paper, the stability of two weakly coupled elastic beams connected vertically by a spring is investigated via the frequency domain method and the multiplier technique. When the two beams have partially local damping, the operator \(\mathcal{A}\) is obtained via variable conversion, and it generating a semigroup is proved, then we obtain that the semigroup is exponentially stable by reduction to absurdity.Discrete-velocity-direction models of BGK-type with minimum entropy. II: Weighted modelshttps://zbmath.org/1544.351732024-11-01T15:51:55.949586Z"Chen, Yihong"https://zbmath.org/authors/?q=ai:chen.yihong"Huang, Qian"https://zbmath.org/authors/?q=ai:huang.qian"Yong, Wen-An"https://zbmath.org/authors/?q=ai:yong.wen-anThe Bathnagar-Gross-Krook (BGK) model is an approximation of the Boltzmann equation, describing the time evolution of a single monoatomic rarefied gas and satisfying the properties: conservation and entropy inequality. The BGK equation for the density function \(f=f(t,x,\xi ,\zeta )\) has the form \(\partial_tf+\xi\cdot \nabla_xf=\tau^{-1} (\mathcal{E}[f]-f)\), where \((x,\xi )\in \mathbb{R}^D\times \mathbb{R}^D\) with \(D=2\) or 3, \(\zeta\in \mathbb{R}^L\) represents the possible internal molecular degrees of freedom, and \(\tau \) is a characteristic collision time. Here, \(\mathcal{E}[f]\) is the local equilibrium state. In the present paper the authors consider a discrete-velocity-direction model (DVDM) with collisions of BGK-type for simulating gas flows, where the molecular motion is confined to some prescribed directions but the speed is still a continuous variable in each direction. The BGK-DVDM is improved in two aspects. First, the internal molecular degrees of freedom is included so that more realistic fluid properties can be realized. The authors introduce a weighted function in each orientation when recovering the macroscopic parameters, as opposed to the previous treatment. For the new weighted DVDM, the established properties of the well-behaved discrete equilibrium still hold. The DVDM is considered under requirement that the molecule transport is limited to \(N\) prescribed directions \(\{\boldsymbol{l}_m\}_{m=1}^{N}\) with each \(\boldsymbol{l}_m\) located on the unit sphere \(\mathbb{S}^{D-1}\), but the velocity magnitude \(\xi\in \mathbb{R}\) in each direction remains continuous. There is a pair of requirements: 1) \((\boldsymbol{l}_1,\dots, \boldsymbol{l}_N)\in \mathbb{R}^{D\times N}\) is of rank \(D\) and therefore \(N\geq D\); 2) Each direction \(\boldsymbol{l}_m\) and its opposite \(-\boldsymbol{l}_m\) belong to \(S_m\subset\mathbb{S}^{D-1}\), where the quantities \(S_m\) constitute a disjoint partition of the unit sphere \(\mathbb{S}^{D-1}= \bigcup_{m=1}^{N}S_m\). Each \(S_m\) has the same measure. Note that the distribution \(f\) is replaced by \(N\) distributions \(\{f_m(t,x,\xi ,\zeta )\}_{m=1}^{N}\) with \(\xi\in\mathbb{R}\) and \(\zeta\in\mathbb{R}^L\). The transport velocity for \(f_m\) is \(\xi\boldsymbol{l}_m\), and the governing equation for \(f_m\) becomes \(\partial_tf_m+\xi\boldsymbol{l}_m\cdot \nabla_xf_m = \tau^{-1} (\mathcal{E}_m-f_m)\) \((m=1,\dots,N)\) with the local equilibriums \(\mathcal{E}_m\). The authors introduce a weighted function in each orientation when recovering the macroscopic parameters. With the weighted DVDM, the authors consider three submodels by incorporating the discrete velocity method, the Gaussian-extended quadrature method of moments and the Hermite spectral method in each direction. It seems the stated spatial-time submodels are multidimensional versions corresponding to the three approaches. Some numerical tests with a series of 1-D and 2-D flow problems show the efficiency of the weighted DVDM.
For Part I see [ibid. 95, No. 3, Paper No. 80, 29 p. (2023; Zbl 1515.65235)].
Reviewer: Dimitar A. Kolev (Sofia)Dissipative soliton resonance: adiabatic theory and thermodynamicshttps://zbmath.org/1544.351742024-11-01T15:51:55.949586Z"Kalashnikov, Vladimir L."https://zbmath.org/authors/?q=ai:kalashnikov.vladimir-l"Rudenkov, Alexander"https://zbmath.org/authors/?q=ai:rudenkov.alexander"Sorokin, Evgeni"https://zbmath.org/authors/?q=ai:sorokin.evgeni"Sorokina, Irina T."https://zbmath.org/authors/?q=ai:sorokina.irina-tSummary: We present the adiabatic theory of dissipative solitons (DS) of complex cubic-quintic nonlinear Ginzburg-Landau equation (CQGLE). Solutions in the closed analytical form in the spectral domain have the shape of Rayleigh-Jeans distribution for a positive (normal) dispersion. The DS parametric space forms a two-dimensional (or three-dimensional for the complex quintic nonlinearity) master diagram connecting the DS energy and a universal parameter formed by the ratio of four real and imaginary coefficients for dissipative and non-dissipative terms in CQGLE. The concept of dissipative soliton resonance (DSR) is formulated in terms of the master diagram, and the main signatures of transition to DSR are demonstrated and experimentally verified. We show a close analogy between DS and incoherent (semicoherent) solitons with an ensemble of quasi-particles confined by a collective potential. It allows applying the thermodynamical approach to DS and deriving the conditions for the DS energy scalability.On the Vlasov-Poisson-Boltzmann limit of the Vlasov-Maxwell-Boltzmann systemhttps://zbmath.org/1544.351752024-11-01T15:51:55.949586Z"Jiang, Ning"https://zbmath.org/authors/?q=ai:jiang.ning"Lei, Yuanjie"https://zbmath.org/authors/?q=ai:lei.yuanjie"Zhao, Huijiang"https://zbmath.org/authors/?q=ai:zhao.huijiangSummary: For the whole range of cutoff intermolecular interactions, we give a rigorous mathematical justification of the limit from the Vlasov-Maxwell-Boltzmann system to the Vlasov-Poisson-Boltzmann system as the light speed tends to infinity. Such a limit is shown to hold global-in-time in the perturbative framework. The key point is to deduce certain \textit{a priori} estimates which are independent of the light speed and the main difficulty is due to the degeneracy of the dissipative effect of the electromagnetic field for large light speed.Volterra-Prabhakar function of distributed order and some applicationshttps://zbmath.org/1544.351762024-11-01T15:51:55.949586Z"Górska, K."https://zbmath.org/authors/?q=ai:gorska.katarzyna"Pietrzak, T."https://zbmath.org/authors/?q=ai:pietrzak.t"Sandev, T."https://zbmath.org/authors/?q=ai:sandev.trifce"Tomovski, Ž."https://zbmath.org/authors/?q=ai:tomovski.zivoradSummary: The paper studies the exact solution of two kinds of generalized Fokker-Planck equations in which the integral kernels are given either by the distributed order function \(k_1 ( t ) = \int_0^1 t^{- \mu} / \varGamma (1 - \mu) \operatorname{d} \mu\) or the distributed order Prabhakar function \(k_2 ( \alpha , \gamma ; \lambda ; t ) = \int_0^1 e_{\alpha , 1 - \mu}^{- \gamma} (\lambda ; t ) \operatorname{d} \mu \), where the Prabhakar function is denoted as \(e_{\alpha , 1 - \mu}^{- \gamma} ( \lambda ; t )\). Both of these integral kernels can be called the fading memory functions and are the Stieltjes functions. It is also shown that their Stieltjes character is enough to ensure the non-negativity of the mean square values and higher even moments. The odd moments vanish. Thus, the solution of generalized Fokker-Planck equations can be called the probability density functions. We introduce also the Volterra-Prabhakar function and its generalization which are involved in the definition of \(k_2 ( \alpha , \gamma ; \lambda ; t )\) and generated by it the probability density function \(p_2 ( x , t )\).A KdV-SIR equation and its analytical solutions for solitary epidemic waveshttps://zbmath.org/1544.351772024-11-01T15:51:55.949586Z"Paxson, Wei"https://zbmath.org/authors/?q=ai:paxson.wei"Shen, Bo-Wen"https://zbmath.org/authors/?q=ai:shen.bo-wen(no abstract)The cortical V1 transform as a heterogeneous Poisson problemhttps://zbmath.org/1544.351782024-11-01T15:51:55.949586Z"Sarti, Alessandro"https://zbmath.org/authors/?q=ai:sarti.alessandro"Galeotti, Mattia"https://zbmath.org/authors/?q=ai:galeotti.mattia"Citti, Giovanna"https://zbmath.org/authors/?q=ai:citti.giovannaThe authors prove that the distribution of cells in the primary visual cortex is sufficient to reconstruct the perceived image without additional constraints. They focus on the perceptual phenomena of lightness and color constancy. The main model of cortical transform as a heterogeneous Poisson is presented. By employing a steepest descent method, the authors prove the existence of a weak solution of this problem. Then the notion of the \(H\)-convergence is introduced, and the proof of convergence of the heterogeneous problem to the homogeneous one is also given. Numerical results are finally addressed and discussed.
Reviewer: Rodica Luca (Iaşi)Boundary output tracking for a flexible beam with tip payload and boundary nonlinear disturbancehttps://zbmath.org/1544.351792024-11-01T15:51:55.949586Z"Mei, Zhan-Dong"https://zbmath.org/authors/?q=ai:mei.zhandong"Arshad, Hizba"https://zbmath.org/authors/?q=ai:arshad.hizba(no abstract)On Hamilton-Jacobi PDEs and image denoising models with certain nonadditive noisehttps://zbmath.org/1544.351802024-11-01T15:51:55.949586Z"Darbon, Jérôme"https://zbmath.org/authors/?q=ai:darbon.jerome"Meng, Tingwei"https://zbmath.org/authors/?q=ai:meng.tingwei"Resmerita, Elena"https://zbmath.org/authors/?q=ai:resmerita.elenaThe paper is concerned with the variational model for denoising problems with nonadditive noise
\[
\displaystyle\min_{v\in\mathrm{int}\,\mathrm{dom}\, H^*}\left\{J^*(\nabla H^*(v))+tD_{H^*}\left(\displaystyle\frac{x}{t},v\right)\right\},\tag{1}
\]
where \(H^*\) and \(J^*\) are the Legendre-Fenchel transforms of convex functions \(H\) and \(J\), respectively, and \(D_{H^*}\left(\displaystyle\frac{x}{t},v\right)\) is the Bregman distance with respect to the function \(H^*\). The authors provide the novel connections between problem \((1)\) and the variational model for additive noise
\[
\displaystyle\min_{v\in\mathbb{R}^n}\left\{J(x-tv)+tH^*(v)\right\}.\tag{2}
\]
They prove a generalized Moreau identity, that is used to show the equivalence of problems \((1)\) and \((2)\) under certain assumptions. The connection of the variational models and Hamilton-Jacobi partial differential equations is established, together with some properties of the H-J PDEs. The authors prove also an asymptotic result of the variational models, and the convergence of the minimizer as the parameter \(t\) goes to infinity under some assumptions. Robust algorithms for the non-convex problem \((1)\) with Poisson and multiplicative noise, and various examples are finally addressed.
Reviewer: Rodica Luca (Iaşi)Highest cusped waves for the fractional KdV equationshttps://zbmath.org/1544.351882024-11-01T15:51:55.949586Z"Dahne, Joel"https://zbmath.org/authors/?q=ai:dahne.joelSummary: In this paper we prove the existence of highest, cusped, traveling wave solutions for the fractional KdV equations \(f_t + ff_x = |D|^\alpha f_x\) for all \(\alpha \in (-1, 0)\) and give their exact leading asymptotic behavior at zero. The proof combines careful asymptotic analysis and a computer-assisted approach.\(q\)-homotopy analysis method for time-fractional Newell-Whitehead equation and time-fractional generalized Hirota-Satsuma coupled KdV systemhttps://zbmath.org/1544.351922024-11-01T15:51:55.949586Z"Liu, Di"https://zbmath.org/authors/?q=ai:liu.di"Gu, Qiongya"https://zbmath.org/authors/?q=ai:gu.qiongya"Wang, Lizhen"https://zbmath.org/authors/?q=ai:wang.lizhen(no abstract)Invariant distributions and the transport twistor space of closed surfaceshttps://zbmath.org/1544.370232024-11-01T15:51:55.949586Z"Bohr, Jan"https://zbmath.org/authors/?q=ai:bohr.jan"Lefeuvre, Thibault"https://zbmath.org/authors/?q=ai:lefeuvre.thibault"Paternain, Gabriel P."https://zbmath.org/authors/?q=ai:paternain.gabriel-pSummary: We study transport equations on the unit tangent bundle of a closed oriented Riemannian surface and their links to the \textit{transport twistor space} of the surface (a complex surface naturally tailored to the geodesic vector field). We show that fibrewise holomorphic distributions invariant under the geodesic flow -- which play an important role in tensor tomography on surfaces -- form a \textit{unital algebra}, that is, multiplication of such distributions is well defined and continuous. We also exhibit a natural bijective correspondence between fibrewise holomorphic invariant distributions and genuine holomorphic functions on twistor space with polynomial blowup on the boundary of the twistor space. Additionally, when the surface is Anosov, we classify holomorphic line bundles over twistor space which are smooth up to the boundary. As a byproduct of our analysis, we obtain a quantitative version of a result of \textit{L. Flaminio} [C. R. Acad. Sci., Paris, Sér. I 315, No. 6, 735--738 (1992; Zbl 0772.58046)]
asserting that invariant distributions of the geodesic flow of a positively curved metric on \(\mathbb{S}^2\) are determined by their zeroth and first Fourier modes.
{\copyright} 2024 The Authors. \textit{Journal of the London Mathematical Society} is copyright {\copyright} London Mathematical Society.Deformation conjecture: deforming lower dimensional integrable systems to higher dimensional ones by using conservation lawshttps://zbmath.org/1544.370502024-11-01T15:51:55.949586Z"Lou, S. Y."https://zbmath.org/authors/?q=ai:lou.senyue"Hao, Xia-zhi"https://zbmath.org/authors/?q=ai:hao.xiazhi"Jia, Man"https://zbmath.org/authors/?q=ai:jia.man.1Summary: Utilizing some conservation laws of \((1+1)\)-dimensional integrable local evolution systems, it is conjectured that higher dimensional integrable equations may be regularly constructed by a deformation algorithm. The algorithm can be applied to Lax pairs and higher order flows. In other words, if the original lower dimensional model is Lax integrable (possesses Lax pairs) and symmetry integrable (possesses infinitely many higher order symmetries and/or infinitely many conservation laws), then the deformed higher order systems are also Lax integrable and symmetry integrable. For concreteness, the deformation algorithm is applied to the usual \((1+1)\)-dimensional Korteweg-de Vries (KdV) equation and the \((1+1)\)-dimensional Ablowitz-Kaup-Newell-Segur (AKNS) system (including nonlinear Schrödinger (NLS) equation as a special example). It is interesting that the deformed \((3+1)\)-dimensional KdV equation is also an extension of the \((1+1)\)-dimensional Harry-Dym (HD) type equations which are reciprocal links of the \((1+1)\)-dimensional KdV equation. The Lax pairs of the \((3 + 1)\)-dimensional KdV-HD system and the \((2 + 1)\)-dimensional AKNS system are explicitly given. The higher order symmetries, i.e., the whole \((3 + 1)\)-dimensional KdV-HD hierarchy, are also explicitly obtained via the deformation algorithm. The single soliton solution of the \((3 + 1)\)-dimensional KdV-HD equation is implicitly given. Because of the effects of the deformation, the symmetric soliton shape of the usual KdV equation is no longer conserved and deformed to be asymmetric and/or multi-valued. The deformation conjecture holds for all the known \((1+1)\)-dimensional integrable local evolution systems that have been checked, and we have not yet found any counter-example so far. The introduction of a large number of \((D + 1)\)-dimensional integrable systems of this paper explores a serious challenge to all mathematicians and theoretical physicists because the traditional methods are no longer directly valid to solve these integrable equations.A combined Liouville integrable hierarchy associated with a fourth-order matrix spectral problemhttps://zbmath.org/1544.370512024-11-01T15:51:55.949586Z"Ma, Wen-Xiu"https://zbmath.org/authors/?q=ai:ma.wen-xiu|ma.wenxiu(no abstract)On the Poisson structure and action-angle variables for the complex modified Korteweg-de Vries equationhttps://zbmath.org/1544.370532024-11-01T15:51:55.949586Z"Yin, Zhe-Yong"https://zbmath.org/authors/?q=ai:yin.zhe-yong"Tian, Shou-Fu"https://zbmath.org/authors/?q=ai:tian.shoufuSummary: In this paper, we employ the inverse scattering approach to study the Poisson structure and action-angle variables of the complex modified Korteweg-de Vries (cmKdV) equation. We first derive the cmKdV equation via the principle of variation. Then, we successfully obtain the Poisson brackets for the scattering data of the equation. Furthermore, the action-angle variables are expressed in terms of the scattering data. Interestingly, our results show that the coordinate expression and the spectral parameter expression of the Hamiltonian can be related by the conservation laws.The Sasa-Satsuma equation with high-order discrete spectra in space-time solitonic regions: soliton resolution via the mixed \(\bar{\partial}\)-Riemann-Hilbert problemhttps://zbmath.org/1544.370542024-11-01T15:51:55.949586Z"Zhang, Minghe"https://zbmath.org/authors/?q=ai:zhang.minghe"Yan, Zhenya"https://zbmath.org/authors/?q=ai:yan.zhenya(no abstract)On a new proof of the Okuyama-Sakai conjecturehttps://zbmath.org/1544.370552024-11-01T15:51:55.949586Z"Yang, Di"https://zbmath.org/authors/?q=ai:yang.di"Zhang, Qingsheng"https://zbmath.org/authors/?q=ai:zhang.qingshengSummary: \textit{K. Okuyama} and \textit{K. Sakai} [J. High Energy Phys. 2020, No. 10, Paper No. 160, 36 p. (2020; Zbl 1456.83116)]
gave a conjectural equality for the higher genus generalized Brézin-Gross-Witten (BGW) free energies. In a recent work [the authors, ``On the Hodge-BGW correspondence'', Preprint, \url{arXiv:2112.12736}], we established the Hodge-BGW correspondence on the relationship between certain special cubic Hodge integrals and the generalized BGW correlators, and a proof of the Okuyama-Sakai conjecture was also given \textit{ibid}. In this paper, we give a new proof of the Okuyama-Sakai conjecture by a further application of the Dubrovin-Zhang theory for the KdV hierarchy.Long-time asymptotics of solution to the coupled Hirota system with \(4 \times 4\) Lax pairhttps://zbmath.org/1544.370592024-11-01T15:51:55.949586Z"Liu, Nan"https://zbmath.org/authors/?q=ai:liu.nanSummary: We study the Cauchy problem for the integrable coupled Hirota system with a \(4 \times 4\) Lax pair on the line with decaying initial data. By deriving a Riemann-Hilbert representation for the solution, we compute the precise leading-order terms for long-time asymptotics based on the nonlinear steepest descent arguments.Long time stability for the derivative nonlinear Schrödinger equationhttps://zbmath.org/1544.370602024-11-01T15:51:55.949586Z"Liu, Jianjun"https://zbmath.org/authors/?q=ai:liu.jianjun|liu.jianjun.1"Xiang, Duohui"https://zbmath.org/authors/?q=ai:xiang.duohuiSummary: In this paper, we consider the long time dynamics of the solutions of the derivative nonlinear Schrödinger equation on one dimensional torus without external parameters. By using rational normal form, we prove the long time stability for generic small initial data.Dynamic analysis of a new financial system with diffusion effect and two delayshttps://zbmath.org/1544.371032024-11-01T15:51:55.949586Z"Wu, Huiming"https://zbmath.org/authors/?q=ai:wu.huiming"Jiang, Zhichao"https://zbmath.org/authors/?q=ai:jiang.zhichao"Wu, Xiaoxue"https://zbmath.org/authors/?q=ai:wu.xiaoxue(no abstract)Ground states for DNLS equation with periodic or asymptotically periodic potentialhttps://zbmath.org/1544.390022024-11-01T15:51:55.949586Z"Chen, Peng"https://zbmath.org/authors/?q=ai:chen.peng.4|chen.peng.1"Meng, Li"https://zbmath.org/authors/?q=ai:meng.li"Tang, Xianhua"https://zbmath.org/authors/?q=ai:tang.xian-huaSummary: We study the existence of ground state solutions for a class of discrete nonlinear Schrödinger equation with a sign-changing potential which is periodic or asymptotically periodic. The resulting problem engages two major difficulties: one is that the associated functional is strongly indefinite, the second is that, due to the asymptotically periodic assumption, the associated functional loses the \(\mathbb{Z}\)-translation invariance, and many effective methods for periodic problems cannot be applied to asymptotically periodic ones. These enables us to develop a direct approach to find ground state solutions with asymptotically periodic potential. Two types of ground state solutions are obtained with some new super-quadratic conditions on nonlinearity which are weaker that some well-known ones. Moreover, our conditions can also be used to significantly improve the well-known results of the corresponding continuous nonlinear Schrödinger equation.Well-posed nonlinear problems. A study of mathematical models of contacthttps://zbmath.org/1544.470012024-11-01T15:51:55.949586Z"Sofonea, Mircea"https://zbmath.org/authors/?q=ai:sofonea.mircea-t|sofonea.mirceaThis book aims at drawing a link between the theory of well-posedness for various types of problems (i.e., differential problems, variational inequalities, split vs. dual problems, etc.) and the theory of modelling for contact mechanics. In particular, the author provides a huge amount of specific examples where he uses the theoretic results that he built. The book is divided in three parts and several chapters.
Part I is dedicated to the theory of well-posedness. Chapter~1 covers some preliminaries as well as some classical definition of well-posedness (namely, Tykhonov's, Levitin-Polyak's, and Hadamard's). Chapter~2 presents the concept of Tykhonov triple (which was introduced by the author and collaborators in the last decade) and provides some basic results (in particular, it is shown that this notion extends all the previous ones presented in Chapter~1).
Part II consists of five chapters, all of them dedicated to examples where the new concept of well-posedness in the sense of Tykhonov triple is demonstrated.
Part II treats contact problems. Chapter~8 covers some preliminaries for the theory of contact mechanics and its mathematical modelling. Chapters 9 and 10 consist of examples of the application of well-posedness in the sense of Tykhonov triple to various, very specific problems.
The author also includes a final chapter of bibliographical notes.
Reviewer: Davide Buoso (Alessandria)Variational-hemivariational system for contaminant convection-reaction-diffusion model of recovered fracturing fluidhttps://zbmath.org/1544.490042024-11-01T15:51:55.949586Z"Cen, Jinxia"https://zbmath.org/authors/?q=ai:cen.jinxia"Migórski, Stanisław"https://zbmath.org/authors/?q=ai:migorski.stanislaw"Yao, Jen-Chih"https://zbmath.org/authors/?q=ai:yao.jen-chih"Zeng, Shengda"https://zbmath.org/authors/?q=ai:zeng.shengdaSummary: This work is devoted to study the convection-reaction-diffusion behavior of contaminant in the recovered fracturing fluid which flows in the wellbore from shale gas reservoir. First, we apply various constitutive laws for generalized non-Newtonian fluids, diffusion principles, and friction relations to formulate the recovered fracturing fluid model. The latter is a partial differential system composed of a nonlinear and nonsmooth stationary incompressible Navier-Stokes equation with a multivalued friction boundary condition, and a nonlinear convection-reaction-diffusion equation with mixed Neumann boundary conditions. Then, we provide the weak formulation of the fluid model which is a hemivariational inequality driven by a nonlinear variational equation. We establish existence of solutions to the recovered fracturing fluid model via a surjectivity theorem for multivalued operators combined with an alternative iterative method and elements of nonsmooth analysis.Differential equations for studies in computational electrophysiologyhttps://zbmath.org/1544.650032024-11-01T15:51:55.949586Z"Jæger, Karoline Horgmo"https://zbmath.org/authors/?q=ai:jaeger.karoline-horgmo"Tveito, Aslak"https://zbmath.org/authors/?q=ai:tveito.aslakPublisher's description: This open access text aims at giving you the simplest possible introduction to differential equations that are used in models of electrophysiology. It covers models at several spatial and temporal scales with associated numerical methods. The text demonstrates that a very limited number of fundamental techniques can be used to define numerical methods for equations ranging from ridiculously simple to extremely complex systems of partial differential equations. Every method is implemented in Matlab and the codes are freely available online. By using these codes, the reader becomes familiar with classical models of electrophysiology, like the cable equation, the monodomain model, and the bidomain model. But modern models that have just started to gain attention in the field of computational electrophysiology are also presented. If you just want to read \textit{one} \textit{book}, it should probably not be this one, but if you want a simple introduction to a complex field, it is worth considering the present text.Structure-preserving numerical approximations for a port-Hamiltonian formulation of the non-isothermal Euler equationshttps://zbmath.org/1544.650072024-11-01T15:51:55.949586Z"Hauschild, Sarah-Alexa"https://zbmath.org/authors/?q=ai:hauschild.sarah-alexaPublisher's description: In this thesis we introduce infinite dimensional port-Hamiltonian formulations of a model library based on the compressible non-isothermal Euler equations to model pipe flow with temperature-dependence.
Additionally, we set up the underlying Stokes-Dirac structures and deduce the boundary port variables. Following that, we adapt the structure-preserving semi-discretization for the isothermal Euler equations to the non-isothermal case. As these systems are highly non-linear we use the extended group finite element method to make the non-linearities easily manageable during model order and complexity reduction. These two procedures are necessary when simulating large networks of pipes in reasonable amounts of time. Thus, we deduce a structure-preserving model order reduction procedure for the single pipe system.
Furthermore, we compare two complexity reduction procedures, i.e., the discrete empirical interpolation method and an empirical quadrature based ansatz, which is even structure-preserving. Finally, we introduce coupling conditions into our port-Hamiltonian formulations, such that the structure of the single pipes is preserved and the whole network system is port-Hamiltonian itself. As the port-Hamiltonian structure is preserved during coupling the numerical methods developed for the single pipe systems can be easily applied to the network case. Academic numerical examples will support our analytical findings.Finite difference methods for nonlinear evolution equationshttps://zbmath.org/1544.650082024-11-01T15:51:55.949586Z"Sun, Zhi-Zhong"https://zbmath.org/authors/?q=ai:sun.zhizhong"Zhang, Qifeng"https://zbmath.org/authors/?q=ai:zhang.qifeng"Gao, Guang-hua"https://zbmath.org/authors/?q=ai:gao.guanghuaThe book gives a wide overview on the solution of nonlinear evolution equations by means of finite difference methods and their numerical analysis. The book contains twelve chapters. Each of these chapters is devoted to a special nonlinear equation. These equations are the Fisher equation, the Burgers' equation; the regularized long-wave equation, the Korteweg-de Vries equation, the Camassa-Holm equation, the one-dimensional Schrödinger equation, the Kuramoto-Tsuzuki equation, the one-dimensional Zakharov equation, the Ginzburg-Landau equation, the Cahn-Hilliard equation, the epitaxial growth model, and the phase field crystal model. For each equation several finite difference schemes are introduced. The conservation, the boundedness, the existence, and the uniqueness of the solution of most of these finite difference schemes are analyzed. The convergence of the solution of all schemes is proved. At the end of each chapter, numerical experiments are presented which illustrate the convergence behavior of the finite difference schemes described before. The book is well suited for all who wants to learn how nonlinear evolution equations can be solved numerically.
Reviewer: Michael Jung (Dresden)Construction and analysis of structure-preserving numerical algorithm for two-dimensional damped nonlinear space fractional Schrödinger equationhttps://zbmath.org/1544.651292024-11-01T15:51:55.949586Z"Ding, Hengfei"https://zbmath.org/authors/?q=ai:ding.hengfei"Qu, Haidong"https://zbmath.org/authors/?q=ai:qu.haidong"Yi, Qian"https://zbmath.org/authors/?q=ai:yi.qianSummary: In this paper, we present a novel high-order structure-preserving numerical scheme for solving the damped nonlinear space fractional Schrödinger equation (DNSFSE) in two spatial dimensions. The main idea of constructing new algorithm consists of two parts. Firstly, we introduce an auxiliary exponential variable to transform the original DNSFSE into a modified one. The modified DNSFSE subjects to the conservation of mass and energy, which is crucial to develop structure-preserving numerical schemes. Secondly, we construct a high-order numerical differential formula to approximate the Riesz derivative in space, which contributes to a semi-discrete difference scheme for the modified DNSFSE. Combining the semi-discrete scheme with the variant Crank-Nicolson method in time, we can obtain the fully-discrete difference scheme for solving the modified DNSFSE. The advantage of the proposed scheme is that a fourth-order convergent accuracy can be achieved in space while maintaining the conservation of mass and energy. Subsequently, we conduct a detailed study on the boundedness, uniqueness, and convergence of solution for fully-discrete scheme. Furthermore, an improved efficient iterative algorithm is proposed for the fully-discrete scheme, which has the advantage of maintaining the same convergence order as the original difference scheme. Finally, extensive numerical results are reported to further verify the correctness of theoretical analysis and the effectiveness of the proposed numerical algorithm.A second order convergent difference scheme for the initial-boundary value problem of Rosenau-Burgers equationhttps://zbmath.org/1544.651302024-11-01T15:51:55.949586Z"Dong, Sitong"https://zbmath.org/authors/?q=ai:dong.sitong"Zhang, Xin"https://zbmath.org/authors/?q=ai:zhang.xin.43|zhang.xin.34|zhang.xin.47|zhang.xin.7|zhang.xin.24|zhang.xin.32|zhang.xin.40|zhang.xin.25|zhang.xin.35|zhang.xin.23|zhang.xin.30|zhang.xin.8|zhang.xin.49|zhang.xin.33|zhang.xin.13|zhang.xin.41|zhang.xin.46|zhang.xin.53"Jin, Yuanfeng"https://zbmath.org/authors/?q=ai:jin.yuanfengSummary: We construct a two-level implicit nonlinear finite difference scheme for the initial boundary value problem of Rosenau-Burgers equation based on the method of order reduction. We discuss conservation, unique solvability, and convergence for the difference scheme. The new scheme is shown to be second-order convergent in time and space. Finally, numerical simulations illustrate our theoretical analysis.Compact local structure-preserving algorithms for the nonlinear Schrödinger equation with wave operatorhttps://zbmath.org/1544.651342024-11-01T15:51:55.949586Z"Huang, Langyang"https://zbmath.org/authors/?q=ai:huang.langyang"Tian, Zhaowei"https://zbmath.org/authors/?q=ai:tian.zhaowei"Cai, Yaoxiong"https://zbmath.org/authors/?q=ai:cai.yaoxiongSummary: Combining the compact method with the structure-preserving algorithm, we propose a compact local energy-preserving scheme and a compact local momentum-preserving scheme for the nonlinear Schrödinger equation with wave operator (NSEW). The convergence rates of both schemes are \(O \left(h^4 + \tau^2\right)\). The discrete local conservative properties of the presented schemes are derived theoretically. Numerical experiments are carried out to demonstrate the convergence order and local conservation laws of the developed algorithms.Numerical methods for fractional Fokker-Planck equation with multiplicative Marcus Lévy noiseshttps://zbmath.org/1544.651402024-11-01T15:51:55.949586Z"Shang, Wenpeng"https://zbmath.org/authors/?q=ai:shang.wenpeng"Cheng, Xiujun"https://zbmath.org/authors/?q=ai:cheng.xiujun"Li, Xiaofan"https://zbmath.org/authors/?q=ai:li.xiaofan"Wang, Xiao"https://zbmath.org/authors/?q=ai:wang.xiao.5Summary: The Fokker-Planck equation (FPE) is an important deterministic tool for investigating stochastic dynamical systems. In this paper, we consider the space-time fractional FPE driven by multiplicative Marcus Lévy noises. Efficient numerical schemes are presented to solve the equations. Stability and convergence of the methods are also discussed. We give some numerical experiments to validate our schemes, and examine the effects of parameters on solutions. Additionally, we analyze the maximal likely trajectories and the critical time for the change of the most probability location.Transformed model reduction for partial differential equations with sharp inner layershttps://zbmath.org/1544.651422024-11-01T15:51:55.949586Z"Tang, Tianyou"https://zbmath.org/authors/?q=ai:tang.tianyou"Xu, Xianmin"https://zbmath.org/authors/?q=ai:xu.xianmin.1|xu.xianminSummary: Small parameters in partial differential equations can give rise to solutions with sharp inner layers that evolve over time. However, the standard model reduction method becomes inefficient when applied to these problems due to the slow decaying Kolmogorov \(N\)-width of the solution manifold. To address this issue, a natural approach is to transform the equation in such a way that the transformed solution manifold exhibits a fast decaying Kolmogorov \(N\)-width. In this paper, we focus on the Allen-Cahn equation as a model problem. We employ asymptotic analysis to identify slow variables and perform a transformation of the partial differential equations accordingly. Subsequently, we apply the proper orthogonal decomposition method and a QR discrete empirical interpolation method (qDEIM) technique to the transformed equation with the slow variables. Numerical experiments demonstrate that the new model reduction method yields significantly improved results compared to direct model reduction applied to the original equation. Furthermore, this approach can be extended to other equations, such as the convection equation and the Burgers equation.Linearized decoupled mass and energy conservation CN Galerkin FEM for the coupled nonlinear Schrödinger systemhttps://zbmath.org/1544.651742024-11-01T15:51:55.949586Z"Shi, Dongyang"https://zbmath.org/authors/?q=ai:shi.dongyang"Qi, Zhenqi"https://zbmath.org/authors/?q=ai:qi.zhenqiSummary: In this paper, a linearized decoupled mass and energy conservation Crank-Nicolson (CN) fully-discrete scheme is proposed for the coupled nonlinear Schrödinger (CNLS) system with the conforming bilinear Galerkin finite element method (FEM), and the unconditional supercloseness and superconvergence error estimates in \(H^1\)-norm are deduced rigorously. Firstly, with the aid of the popular time-space splitting technique, that is, by introducing a suitable time discrete system, the error is divided into two parts, the time error and spatial error, the boundedness of numerical solution in \(L^{\infty}\)-norm is derived strictly without any constraint between the mesh size \(h\) and the time step \(\tau\). Then, thanks to the high accuracy result between the interpolation and Ritz projection, the unconditional superclose error estimate is obtained, and the corresponding unconditional superconvergence result is acquired through the interpolation post-processing technique. At last, some numerical results are supplied to verify the theoretical analysis.Structure preserving FEM for the perturbed wave equation of quantum mechanicshttps://zbmath.org/1544.651752024-11-01T15:51:55.949586Z"Wang, Junjun"https://zbmath.org/authors/?q=ai:wang.junjun"Chen, Rui"https://zbmath.org/authors/?q=ai:chen.rui"Ma, Wenjing"https://zbmath.org/authors/?q=ai:ma.wenjing"Zhao, Weijie"https://zbmath.org/authors/?q=ai:zhao.weijieSummary: The construction and analysis of structure-preserving finite element method (FEM) for computing the perturbed wave equation of quantum mechanics are demonstrated. Firstly, a new fully discrete system is built and proved conservative in the sense of the energy. Meanwhile, the boundedness of the numerical solution is derived. Secondly, the existence and uniqueness of the solution are obtained with the help of the Brouwer fixed-point theorem and some special splitting technique. Thirdly, we provide a comprehensive superclose analysis, offering the global superconvergent result. Finally, numerical results are presented to illustrate the theoretical analysis.A phase field method for convective phase change problem preserving maximum bound principlehttps://zbmath.org/1544.651792024-11-01T15:51:55.949586Z"Yao, Hui"https://zbmath.org/authors/?q=ai:yao.huiSummary: Numerical simulations of convective solid-liquid phase change problems have long been a complex problem due to the movement of the solid-liquid interface layer, which leads to a free boundary problem. This work develops a convective phase change heat transfer model based on the phase field method. The governing equations consist of the incompressible Navier-Stokes-Boussinesq equations, the heat transfer equation, and the Allen-Cahn equation. The Navier-Stokes equations are penalised for imposing zero velocity within the solid region. For numerical methods, the mini finite element approach (\texttt{P1b-P1}) is used to solve the momentum equation spatially, the temperature and the phase field are approximated by the elements. In the temporal discretization, the phase field and the temperature are decoupled from the momentum equation by using the finite difference method, forming a solvable linear system. A maximum bound principle for the phase field is derived, coming with an estimation of the tolerance of the time step size, which depends on the temperature range. This estimation guides the time step choice in the simulation. The program is developed within the \texttt{FreeFem++} framework, drawing on our previous work on phase field methods [\textit{H. Yao} and \textit{M. Azaiez}, Comput. Methods Appl. Mech. Eng. 421, Article ID 116794, 20 p. (2024; Zbl 1539.65160)] and a mushy-region method toolbox for heat transfer [\textit{G. Sadaka} et al., Comput. Phys. Commun. 257, Article ID 107492, 26 p. (2020; Zbl 1515.76102)]. The accuracy and effectiveness of the proposed method have been validated through real-world cases of melting and solidification with linear or nonlinear buyangcy force, respectively. The simulation results are in agreement with experiments in references.A fast and accurate coupled meshless algorithm for the 2D/3D Gross-Pitaevskii equations on two GPUshttps://zbmath.org/1544.651842024-11-01T15:51:55.949586Z"Jiang, Tao"https://zbmath.org/authors/?q=ai:jiang.tao.3"Wei, Xiang-Yang"https://zbmath.org/authors/?q=ai:wei.xiangyang"Li, Yue"https://zbmath.org/authors/?q=ai:li.yue"Wang, Deng-Shan"https://zbmath.org/authors/?q=ai:wang.dengshan"Yuan, Jin-Yun"https://zbmath.org/authors/?q=ai:yuan.jinyunSummary: This paper first presents a high-efficient and accurate coupled meshless algorithm for solving the multi-dimensional Gross-Pitaevskii equation (GPE) in unbounded domain, which is implemented on CUDA-program-based two-GPUs cards. The proposed novel high-performance scheme (RDFPM-PML-GPU) is mainly motived by the items below: (a) a reduced-dimensional finite pointset method (RDFPM) is first presented to solve the 2D/3D spatial derivatives in GPE, which has lower calculated amount than the traditional FPM (TFPM) for the derivatives; (b) the perfectly matched layer (PML) technique is adopted to treat the absorbing boundary conditions (ABCs) which is used for the infinite exterior region, and the time-splitting technique is resorted to reduce the computing complexity in PML; (c) a fast parallel algorithm based on CUDA-program is proposed to accelerate the computation in the proposed meshless scheme with local matrix on two GPUs. The numerical convergent rate and advantages of the proposed meshless scheme are demonstrated by solving two examples, which include the comparisons between the proposed RDFPM and TFPM, the merit of easily implemented local refinement point distribution in meshless method, and the merit of PML-ABCs over the zero Dirichlet boundary treatment for unbounded domain. Meanwhile, the high efficiency of the proposed GPU-based parallelization algorithm is tested and discussed by simulating 3D examples, which shows that the speed-up rate is about 500-times of using two-GPUs over a single CPU. Finally, the proposed RDFPM-PML-GPU method is used to predict the long-time evolution of quantum vortex in 2D/3D GPEs describing Bose-Einstein condensates. All the numerical tests show the high-performance and flexible application of the proposed parallel meshless algorithm.Low regularity full error estimates for the cubic nonlinear Schrödinger equationhttps://zbmath.org/1544.651852024-11-01T15:51:55.949586Z"Ji, Lun"https://zbmath.org/authors/?q=ai:ji.lun"Ostermann, Alexander"https://zbmath.org/authors/?q=ai:ostermann.alexander"Rousset, Frédéric"https://zbmath.org/authors/?q=ai:rousset.frederic"Schratz, Katharina"https://zbmath.org/authors/?q=ai:schratz.katharinaSummary: For the numerical solution of the cubic nonlinear Schrödinger equation with periodic boundary conditions, a pseudospectral method in space combined with a filtered Lie splitting scheme in time is considered. This scheme is shown to converge even for initial data with very low regularity. In particular, for data in \(H^s(\mathbb{T}^2)\), where \(s>0\), convergence of order \(\mathcal{O}(\tau^{s/2}+N^{-s})\) is proved in \(L^2\). Here \(\tau\) denotes the time step size and \(N\) the number of Fourier modes considered. The proof of this result is carried out in an abstract framework of discrete Bourgain spaces; the final convergence result, however, is given in \(L^2\). The stated convergence behavior is illustrated by several numerical examples.Adaptive absorbing boundary layer for the nonlinear Schrödinger equationhttps://zbmath.org/1544.651872024-11-01T15:51:55.949586Z"Stimming, Hans Peter"https://zbmath.org/authors/?q=ai:stimming.hans-peter"Wen, Xin"https://zbmath.org/authors/?q=ai:wen.xin"Mauser, Norbert J."https://zbmath.org/authors/?q=ai:mauser.norbert-juliusSummary: We present an adaptive absorbing boundary layer technique for the nonlinear Schrödinger equation that is used in combination with the Time-splitting Fourier spectral method (TSSP) as the discretization for the NLS equations. We propose a new complex absorbing potential (CAP) function based on high order polynomials, with the major improvement that an explicit formula for the coefficients in the potential function is employed for adaptive parameter selection. This formula is obtained by an extension of the analysis in [\textit{R. Kosloff} and \textit{D. Kosloff}, J. Comput. Phys. 63, 363--376 (1986; Zbl 0644.65086)]. We also show that our imaginary potential function is more efficient than what is used in the literature. Numerical examples show that our ansatz is significantly better than existing approaches. We show that our approach can very accurately compute the solutions of the NLS equations in one dimension, including in the case of multi-dominant wave number solutions.A fast and efficient numerical algorithm for the nonlocal conservative Swift-Hohenberg equationhttps://zbmath.org/1544.651882024-11-01T15:51:55.949586Z"Wang, Jingying"https://zbmath.org/authors/?q=ai:wang.jingying"Zhai, Shuying"https://zbmath.org/authors/?q=ai:zhai.shuyingSummary: In this paper, we consider a new Swift-Hohenberg equation, where the total mass of this model is conserved through a nonlocal Lagrange multiplier. Based on the operator splitting method and spectral method, a fast and efficient numerical algorithm is proposed. Three numerical examples in both two and three dimensions are provided to illustrate that the proposed algorithm is a practical, accurate, and efficient simulation tool for the nonlocal Swift-Hohenberg equation.A class of new implicit compact sixth-order approximations for Poisson equations and the estimates of normal derivatives in multi-dimensionshttps://zbmath.org/1544.651952024-11-01T15:51:55.949586Z"Mohanty, R. K."https://zbmath.org/authors/?q=ai:mohanty.ranjan-kumar"Niranjan"https://zbmath.org/authors/?q=ai:niranjan.mahasan|niranjan.u-n|niranjan.k-m|niranjan.mahesan|niranjan.u-c|niranjan.p-k|niranjan.s-p|niranjan.h|niranjan.utkarshSummary: In this piece of work, a family of compact implicit numerical algorithms for \((\partial u/\partial n)\) of order of accuracy six are proposed on a 9- and 19-point compact cell for two- and three- dimensional Poisson equations \(\varDelta^2 u=f\) which are quite often useful in mathematical physics and engineering, where \(\varDelta^2\) is either two or three dimensional Laplacian operator. First, we propose a family of new numerical algorithms of order of accuracy six for the computation of the solution of 2D and 3D Poisson equations on 9- and 27-points compact stencil, respectively. Then with the aid of the numerical solution of \(u\), we propose a new family of compact sixth order implicit numerical algorithms for the estimates of \((\partial u/\partial n)\). The proposed algorithms are free from derivatives of the source functions, which makes our algorithms more efficient for computation. Suitable iteration techniques are used for computation to demonstrate the sixth order convergence of the proposed algorithms. Numerical results are tabulated, confirming the usefulness of the suggested numerical algorithms.Frequency-explicit a posteriori error estimates for discontinuous Galerkin discretizations of Maxwell's equationshttps://zbmath.org/1544.652022024-11-01T15:51:55.949586Z"Chaumont-Frelet, Théophile"https://zbmath.org/authors/?q=ai:chaumont-frelet.theophile"Vega, Patrick"https://zbmath.org/authors/?q=ai:vega.patrickSummary: We propose a new residual-based a posteriori error estimator for discontinuous Galerkin discretizations of time-harmonic Maxwell's equations in first-order form. We establish that the estimator is reliable and efficient, and the dependency of the reliability and efficiency constants on the frequency is analyzed and discussed. The proposed estimates generalize similar results previously obtained for the Helmholtz equation and conforming finite element discretizations of Maxwell's equations. In addition, for the discontinuous Galerkin scheme considered here, we also show that the proposed estimator is asymptotically constant-free for smooth solutions.Efficient high-order space-angle-energy polytopic discontinuous Galerkin finite element methods for linear Boltzmann transporthttps://zbmath.org/1544.652042024-11-01T15:51:55.949586Z"Houston, Paul"https://zbmath.org/authors/?q=ai:houston.paul"Hubbard, Matthew E."https://zbmath.org/authors/?q=ai:hubbard.matthew-e"Radley, Thomas J."https://zbmath.org/authors/?q=ai:radley.thomas-j"Sutton, Oliver J."https://zbmath.org/authors/?q=ai:sutton.oliver-j"Widdowson, Richard S. J."https://zbmath.org/authors/?q=ai:widdowson.richard-s-jSummary: We introduce an \(hp\)-version discontinuous Galerkin finite element method (DGFEM) for the linear Boltzmann transport problem. A key feature of this new method is that, while offering arbitrary order convergence rates, it may be implemented in an almost identical form to standard multigroup discrete ordinates methods, meaning that solutions can be computed efficiently with high accuracy and in parallel within existing software. This method provides a unified discretisation of the space, angle, and energy domains of the underlying integro-differential equation and naturally incorporates both local mesh and local polynomial degree variation within each of these computational domains. Moreover, general polytopic elements can be handled by the method, enabling efficient discretisations of problems posed on complicated spatial geometries. We study the stability and \(hp\)-version a priori error analysis of the proposed method, by deriving suitable \(hp\)-approximation estimates together with a novel inf-sup bound. Numerical experiments highlighting the performance of the method for both polyenergetic and monoenergetic problems are presented.Quadrature-free polytopic discontinuous Galerkin methods for transport problemshttps://zbmath.org/1544.652102024-11-01T15:51:55.949586Z"Radley, Thomas J."https://zbmath.org/authors/?q=ai:radley.thomas-j"Houston, Paul"https://zbmath.org/authors/?q=ai:houston.paul"Hubbard, Matthew E."https://zbmath.org/authors/?q=ai:hubbard.matthew-eSummary: In this article we consider the application of Euler's homogeneous function theorem together with Stokes' theorem to exactly integrate families of polynomial spaces over general polygonal and polyhedral (polytopic) domains in two and three dimensions, respectively. This approach allows for the integrals to be evaluated based on only computing the values of the integrand and its derivatives at the vertices of the polytopic domain, without the need to construct a sub-tessellation of the underlying domain of interest. Here, we present a detailed analysis of the computational complexity of the proposed algorithm and show that this depends on three key factors: the ambient dimension of the underlying polytopic domain; the size of the requested polynomial space to be integrated; and the size of a directed graph related to the polytopic domain. This general approach is then employed to compute the volume integrals arising within the discontinuous Galerkin finite element approximation of the linear transport equation. Numerical experiments are presented which highlight the efficiency of the proposed algorithm when compared to standard quadrature approaches defined on a sub-tessellation of the polytopic elements.Essential perturbation methodshttps://zbmath.org/1544.740012024-11-01T15:51:55.949586Z"Wang, C. Y."https://zbmath.org/authors/?q=ai:wang.chien-yao|wang.chenyong|wang.chuyuan|wang.chenye|wang.chuyang|wang.chiyuan.1|wang.chun-yen|wang.changying|wang.chuang-yun|wang.cheng-yeh|wang.chun-yao|wang.ching-yuan|wang.china-yuan|wang.chih-yu|wang.chengyong|wang.chengyang|wang.chen-ye|wang.chenyan|wang.chunyuan|wang.chaoyu|wang.chenying|wang.chengyao|wang.chengyou|wang.chih-ying|wang.chunyang|wang.chinying|wang.chengyuan|wang.chuanyi.1|wang.changyuan.2|wang.changyou|wang.chaoyue|wang.chenyuan|wang.chenyi|wang.chaoyong|wang.cen-yang|wang.chuangyu|wang.chunyun|wang.chengyuan.2|wang.chung-yang|wang.chaoyang|wang.chuanyou|wang.chuanyi|wang.chunyu|wang.chaoyi|wang.chanyu|wang.chengyi|wang.chunyi|wang.cuiying|wang.chenyang|wang.canyou|wang.cuiyu|wang.congyin|wang.chung-yi|wang.chunyan|wang.chuan-yun|wang.chuyun|wang.cangyuan|wang.chung-yung|wang.congyue|wang.chuying|wang.chuan-yin|wang.chi-yu|wang.canyun|wang.chang-yi|wang.chenyu|wang.chengyue|wang.chuanyong|wang.cuiyun|wang.changyou.1|wang.chuyi|wang.chanyuan|wang.chunyong|wang.chunying|wang.chongyu|wang.chengyuan.1|wang.cun-yun|wang.chuanyuan|wang.changyuan|wang.changyuan.1|wang.caiyuan|wang.chengyu|wang.chenyin|wang.chengying|wang.chih-yi|wang.ching-yun|wang.chung-yue|wang.chien-yi|wang.c-y|wang.cunyou|wang.chunyue|wang.changyun|wang.chongyang.1|wang.ching-yao|wang.caiyun|wang.chih-yueh|wang.chunyin|wang.chuanyu|wang.changyuPublisher's description: This book presents the modeling and scaling of physical problems, which result in normalized perturbation equations. This is followed by solving perturbation problems and evaluating the results. The author refines perturbation methods into simple, understandable elements and avoids unnecessary theorems and proofs. In addition, the results are consolidated and interpreted, and the presented examples are succinct to illustrate the essential techniques. This book is ideal and beneficial for practicing scientists and engineers who need to understand and apply perturbation methods to difficult problems with applications in mathematics, engineering, and biology. Discussions on new perspectives, simpler presentations on convergence, and the expansion of integrals are included.Nonlocal-to-local limit in linearized viscoelasticityhttps://zbmath.org/1544.740122024-11-01T15:51:55.949586Z"Friedrich, Manuel"https://zbmath.org/authors/?q=ai:friedrich.manuel"Seitz, Manuel"https://zbmath.org/authors/?q=ai:seitz.manuel"Stefanelli, Ulisse"https://zbmath.org/authors/?q=ai:stefanelli.ulisseSummary: We study the quasistatic evolution of a linear peridynamic Kelvin-Voigt viscoelastic material. More specifically, we consider the gradient flow of a nonlocal elastic energy with respect to a nonlocal viscous dissipation. Following an evolutionary \(\Gamma\)-convergence approach, we prove that the solutions of the nonlocal problem converge to the solution of the local problem, when the peridynamic horizon tends to 0, that is, in the nonlocal-to-local limit.Linearization and computation for large-strain visco-elasticityhttps://zbmath.org/1544.740132024-11-01T15:51:55.949586Z"Dondl, Patrick"https://zbmath.org/authors/?q=ai:dondl.patrick-w"Jesenko, Martin"https://zbmath.org/authors/?q=ai:jesenko.martin"Kružík, Martin"https://zbmath.org/authors/?q=ai:kruzik.martin"Valdman, Jan"https://zbmath.org/authors/?q=ai:valdman.janTime-semidiscretized problems of nonlinear visco-elasticity in Kelvin-Voigt rheology at high strains are not well-posed due to the non-quasi-convexity of the dissipation functional. To circumvent this drawback, the authors linearize the energy functional in the spirit of [\textit{G. Dal Maso} et al., Set-Valued Anal. 10, No. 2--3, 165--183 (2002; Zbl 1009.74008)] to obtain a quadratic energy contribution that stems from the dissipation and stress of the elastic energy. The resulting functional is not coercive, but, in spite of this, minimizers are shown to exist. The authors perform numerical experiments by comparing the original nonlinear scheme with their proposed linearized scheme for small time steps that show very good quantitative agreement. Moreover, both schemes satisfy an energy balance.
Reviewer: Song Jiang (Beijing)Study of the stability properties for a general shape of damped Euler-Bernoulli beams under linear boundary conditionshttps://zbmath.org/1544.740562024-11-01T15:51:55.949586Z"Isaac, Teya Kouakou Kra"https://zbmath.org/authors/?q=ai:isaac.teya-kouakou-kra"Jean-Marc, Bomisso Gossrin"https://zbmath.org/authors/?q=ai:jean-marc.bomisso-gossrin"Touré, Kidjegbo Augustin"https://zbmath.org/authors/?q=ai:augustin.toure-kidjegbo"Coulibaly, Adama"https://zbmath.org/authors/?q=ai:coulibaly.adama(no abstract)Quasistatic crack growth in elasto-plastic materials with hardening: the antiplane casehttps://zbmath.org/1544.740782024-11-01T15:51:55.949586Z"Dal Maso, Gianni"https://zbmath.org/authors/?q=ai:dal-maso.gianni"Toader, Rodica"https://zbmath.org/authors/?q=ai:toader.rodicaSummary: We study a variational model for crack growth in elasto-plastic materials with hardening in the antiplane case. The main result is the existence of a solution to the initial value problem with prescribed time-dependent boundary conditions.A data-driven framework for evolutionary problems in solid mechanicshttps://zbmath.org/1544.740912024-11-01T15:51:55.949586Z"Poelstra, Klaas"https://zbmath.org/authors/?q=ai:poelstra.klaas-hendrik"Bartel, Thorsten"https://zbmath.org/authors/?q=ai:bartel.thorsten"Schweizer, Ben"https://zbmath.org/authors/?q=ai:schweizer.benSummary: Data-driven schemes introduced a new perspective in elasticity: While certain physical principles are regarded as invariable, material models for the relation between strain and stress are replaced by data clouds of admissible pairs of these variables. A data-driven approach is of particular interest for plasticity problems, since the material modeling is even more unclear in this field. Unfortunately, so far, data-driven approaches to evolutionary problems are much less understood. We try to contribute in this area and propose an evolutionary data-driven scheme. We present a first analysis of the scheme regarding existence and data convergence. Encouraging numerical tests are also included.
{\copyright} 2022 The Authors. \textit{ZAMM - Journal of Applied Mathematics and Mechanics} published by Wiley-VCH GmbH.Mathematical methods and modeling for mixtures of fluids and interface evolutionhttps://zbmath.org/1544.760032024-11-01T15:51:55.949586ZPublisher's description: This volume includes the lecture notes of the three mini-courses that have been given during the CIRM-SMF week entitled Inhomogeneous Flows: Asymptotic Models and Interfaces Evolution that took place at the CIRM from September 23 -- September 27, 2019. The conference followed the themes of the ANR-15-CE40-0011 INFAMIE project and aimed at bringing together experts coming from various branches of mathematical fluid dynamics and interested in inhomogeneous fluids where problems of interfaces occur. Beside the mini-courses, the event comprised 14 plenary talks that were specifically dedicated to inhomogeneous flows.
The mini-courses emphasized, on the one hand, theoretical approaches that proved to be efficient in the study of evolutionary fluid mechanics (maximal regularity issues in the notes of P. Kunstmann with stress on the L1
in time estimates that turn out to be crucial in the study of free boundary problems, and the prospective course by P. Auscher on tent spaces), and, on the other hand, the modeling aspect with the lectures by S. Gavrilyuk devoted to the derivation of models of bi-fluids by means of Hamiltonian principle.
The originality of these texts is that they have been written conjointly by the speaker and young participants from the notes that were taken during the courses.
A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30\% discount from list.
The articles of this volume will be reviewed individually.Homogeneous incompressible Bingham viscoplastic as a limit of bi-viscosity fluidshttps://zbmath.org/1544.760062024-11-01T15:51:55.949586Z"Aberqi, Ahmed"https://zbmath.org/authors/?q=ai:aberqi.ahmed"Aboussi, Wassim"https://zbmath.org/authors/?q=ai:aboussi.wassim"Benkhaldoun, Fayssal"https://zbmath.org/authors/?q=ai:benkhaldoun.fayssal"Bennouna, Jaouad"https://zbmath.org/authors/?q=ai:bennouna.jaouad"Bradji, Abdallah"https://zbmath.org/authors/?q=ai:bradji.abdallahThe authors consider a smooth domain \(\Omega \) in \(\mathbb{R}^{2}\) with Lipschitz boundary and a 2D unsteady flow of an incompressible Bingham fluid which is governed by the following Navier-Stokes system: \(\partial _{t}u+(u\cdot \nabla )u-\nabla \cdot (\tau (Du))+\nabla p=f\), \(\nabla \cdot u=0\) in \(\Omega \times (0,T)\), \(T>0\) finite, where \(u\) is the velocity vector, \(p\) the pressure, \(\tau \) the stress tensor, \(Du\) the strain tensor defined as \(Du=\frac{1}{2}(\nabla u+\nabla u^{t})\), and \(f:T\rightarrow \mathbb{R}^{2}\) represents the external forces. The problem is completed with the initial condition \(u(\cdot ,0)=u_{0}\) in \(\Omega \), and the homogeneous Dirichlet boundary condition \(u=0\) on \(\partial \Omega \times (0,T)\). The Bingham stress-strain constitutive law is defined as: \(\tau (Du)=(2\mu + \frac{\tau _{y}}{\left\vert Du\right\vert })Du\), if \(\left\vert \tau \right\vert >\tau _{y}\), \(Du=0\), if \(\left\vert \tau \right\vert \leq \tau _{y}\), where \(\mu \) is the viscosity, \(\tau _{y}\) is the yield stress and \( \left\vert A\right\vert ^{2}=A:A\). The main result of the paper proves that if \(f\in L^{2}(0,T;V^{\prime })\), \(V^{\prime }\) being the topological dual of \(V=\{v\in H_{0}^{1}(\Omega )\), \(\nabla \cdot v=0\}\), and \(u_{0}\in H=\{v\in L^{2}(\Omega )\), \(\nabla \cdot v=0\), \(v\cdot n\mid _{\partial \Omega }=0\}\), the above Navier-Stokes system for a Bingham fluid has a weak solution such that \(u\in L^{2}(0,T;V)\cap L^{\infty }(0,T;H)\), \( \partial _{t}u\in L^{2}(0,T;V)\), and \(\tau (Du)\in L^{2}(T)\). The proof is based on the construction of an an approximate problem by regularizing the Bingham tensor, with a regularizing tensor given by the bi-viscosity model: \( \tau _{m}(A)=2m\mu A\), if \(\left\vert A\right\vert \leq \gamma _{m}\), \(\tau _{m}(A)(2\mu +\frac{\tau _{y}}{\left\vert A\right\vert })A\), if \(\left\vert A\right\vert >\gamma _{m}\), where \(A\in M^{2\times 2}\) and \(\gamma _{m}= \frac{\tau _{y}}{2\mu (m-1)}\), \(m\geq 2\). Assuming that \(f\in L^{2}(0,T;V^{\prime })\) and \(u_{0}\in H\), the authors prove the existence of at least one weak solution \(u_{m}\) to this approximate problem in the Banach space \(E_{2,2}(V)=\{v\in L^{2}(0,T;V)\), \(\partial _{t}v\in L^{2}(0,T;V^{\prime })\}\). They prove that \(u_{m}\) is continuous from \([0,T] \) into \(H\) and they deduce the existence result from bounds on \(u_{m}\) and on its time and spatial derivatives and passing to the limit on \(m\). The authors finally prove the uniqueness of a solution to the problem under consideration.
Reviewer: Alain Brillard (Riedisheim)Asymptotic behavior and internal stabilization for the micropolar fluid equationshttps://zbmath.org/1544.760072024-11-01T15:51:55.949586Z"Braz e. Silva, P."https://zbmath.org/authors/?q=ai:braz-e-silva.pablo"Loayza, M."https://zbmath.org/authors/?q=ai:loayza.miguel"Rojas-Medar, M. A."https://zbmath.org/authors/?q=ai:rojas-medar.marko-antonioSummary: We are interested in the stability of stationary solutions for the incompressible micropolar fluids system. We prove stability results in the spaces \(L^2(\Omega)\), \(H^1(\Omega)\) and \(H^2(\Omega)\). Moreover, we show that this system can be stabilized (in the \(L^2\)-norm) using feedback controllers acting only on a part of the domain of interest.Effects of homogeneous and heterogeneous chemical features on Oldroyd-B fluid flow between stretching disks with velocity and temperature boundary assumptionshttps://zbmath.org/1544.760092024-11-01T15:51:55.949586Z"Khan, Nargis"https://zbmath.org/authors/?q=ai:khan.nargis"Hashmi, Muhammad Sadiq"https://zbmath.org/authors/?q=ai:hashmi.muhammad-sadiq"Khan, Sami Ullah"https://zbmath.org/authors/?q=ai:khan.sami-ullah"Chaudhry, Faryal"https://zbmath.org/authors/?q=ai:chaudhry.faryal"Tlili, Iskander"https://zbmath.org/authors/?q=ai:tlili.iskander"Shadloo, Mostafa Safdari"https://zbmath.org/authors/?q=ai:shadloo.mostafa-safdariSummary: This research endeavors the rheological features of Oldroyd-B fluid configured by infinite stretching disks in presence of velocity and thermal slip features. Additionally, the effects of homogeneous and heterogeneous chemical features are also considered. The transmuted flow equations are analytically solved with help of the homotopy analysis method (HAM). It is observed that the homogeneous chemical reaction parameter enhances the concentration distribution, while the heterogeneous reaction reduces the concentration profile. With implementations of temperature jump conditions, the heat transfer from the surfaces of both disks can be effectively controlled. The impacts of various dimensionless parameters are elaborated through graphs and tables.Incompressible jet flow past an obstaclehttps://zbmath.org/1544.760172024-11-01T15:51:55.949586Z"Cheng, Jianfeng"https://zbmath.org/authors/?q=ai:cheng.jianfeng"Huang, Jinli"https://zbmath.org/authors/?q=ai:huang.jinliThe authors consider the well-posedness of the incompressible jet flow issuing from a semi-infinitely long nozzle and moving around an obstacle. They obtain that one can find a differential pressure, such that there exists a unique solution to the incompressible jet flow, with the upper free boundary initiating smoothly from the endpoint of the nozzle wall, and the lower free boundary initiating smoothly from the surface of the obstacle. Moreover, the nonexistence of a finite cavity between the cavity boundary and the obstacle is established. Additionally, the authors show the optimal regularity at the detachment of the lower free boundary.
Reviewer: Fatma Gamze Düzgün (Ankara)The vortex patch issue for the generalized Boussinesq systemhttps://zbmath.org/1544.760182024-11-01T15:51:55.949586Z"Maafa, Youssouf"https://zbmath.org/authors/?q=ai:maafa.youssouf"Melkemi, Oussama"https://zbmath.org/authors/?q=ai:melkemi.oussamaThe paper is devoted to an analysis of the Boussinesq system in two space dimensions with a smooth source term describing the buoyancy forces so that this generalizes the stratified Euler equations. The main results include global-in-time existence of solutions and persistence of smoothness of vortex patches.
Reviewer: Piotr Biler (Wrocław)Negative Pell equation and stationary configurations of point vortices on the planehttps://zbmath.org/1544.760192024-11-01T15:51:55.949586Z"Vishnevskaya, A. D."https://zbmath.org/authors/?q=ai:vishnevskaya.a-d"Demina, M. V."https://zbmath.org/authors/?q=ai:demina.maria-vIn this paper, the authors consider the model of point vortices coming from the system of Helmholtz equations. They obtain necessary and sufficient conditions for the existence of infinitely many nonequivalent stationary configurations for a system consisting of two point vortices of intensity \(\Gamma_1\) and an arbitrary number of point vortices of intensity \(\Gamma_2\). Moreover, they show a connection between the negative Diophantine Pell equation and a number of stationary configurations of point vortices on the plane.
Reviewer: Fatma Gamze Düzgün (Ankara)Corrigendum to: ``Non-linear singularity formation for circular vortex sheets''https://zbmath.org/1544.760312024-11-01T15:51:55.949586Z"Murray, Ryan"https://zbmath.org/authors/?q=ai:murray.ryan-w"Wilcox, Galen"https://zbmath.org/authors/?q=ai:wilcox.galenSummary: This paper serves as a short corrigendum to the authors' paper [ibid. 82, No. 1, 81--96 (2024; Zbl 1539.76038)], describing an error in the derivation of the linearized equation. The main theorems remain correct, but they are based upon an incorrect linearization of the Birkhoff-Rott equation and hence are not directly applicable to the original physical problem.Optimal decay rates of the compressible Euler equations with time-dependent damping in \(\mathbb{R}^n\). I: Under-damping casehttps://zbmath.org/1544.760942024-11-01T15:51:55.949586Z"Ji, Shanming"https://zbmath.org/authors/?q=ai:ji.shanming"Mei, Ming"https://zbmath.org/authors/?q=ai:mei.mingSummary: This paper is concerned with the multi-dimensional compressible Euler equations with time-dependent damping of the form \(-\frac{\mu}{(1+t)^{\lambda}}\rho\boldsymbol{u}\) in \(\mathbb{R}^n\), where \(n\geq 2, \mu >0\), and \(\lambda \in [0,1)\). When \(\lambda >0\) is bigger, the damping effect time-asymptotically gets weaker, which is called under-damping. We show the optimal decay estimates of the solutions such that \(\Vert \partial_x^{\alpha} (\rho -1)\Vert_{L^2 (\mathbb{R}^n)}\approx (1+t)^{-\frac{1+\lambda}{2}\left(\frac{n}{2}+|\alpha |\right)}\), and \(\Vert \partial_x^{\alpha}\boldsymbol{u}\Vert_{L^2 (\mathbb{R}^n)}\approx (1+t)^{-\frac{1+\lambda}{2}\left(\frac{n}{2}+|\alpha |\right) -\frac{1-\lambda}{2}}\), and see how the under-damping effect influences the structure of the Euler system. Different from the traditional view that the stronger damping usually makes the solutions decaying faster, here we recognize that the weaker damping with \(0\leq \lambda <1\) enhances the faster decay for the solutions. The adopted approach is the technical Fourier analysis and the Green function method. The main difficulties caused by the time-dependent damping lie in twofold: non-commutativity of the Fourier transform of the linearized operator precludes explicit expression of the fundamental solution; time-dependent evolution implies that the Green matrix \(G(t, s)\) is not translation invariant, i.e., \(G(t,s)\neq G(t-s,0)\). We formulate the exact decay behavior of the Green matrices \(G(t, s)\) with respect to \(t\) and \(s\) for both linear wave equations and linear hyperbolic system, and finally derive the optimal decay rates for the nonlinear Euler system.Global dynamics of large solution for the compressible Navier-Stokes-Korteweg equationshttps://zbmath.org/1544.760952024-11-01T15:51:55.949586Z"Song, Zihao"https://zbmath.org/authors/?q=ai:song.zihaoThis paper studies the singular limit as the capillarity coefficient goes to infinity, and the large-time behavior of solutions to the Cauchy problem for the multi-dimensional Navier-Stokes-Korteweg equations governing the evolution of compressible fluids with capillarity effects. By taking the strong dispersion due to large capillarity coefficient into consideration, the author first establishes the global well-posedness of solutions in the critical Besov space for large initial data when the capillarity coefficient is large enough. Then, the author obtains uniform estimates and incompressible limits as the capillarity coefficient tends to infinity by employing a dissipative-dispersive estimate. Finally, the author exploits both parabolic mechanics and dispersive structure to establish the large-time behavior of the solutions without requiring smallness for initial assumption.
Reviewer: Song Jiang (Beijing)Construction of boundary conditions for Navier-Stokes equations from the moment systemhttps://zbmath.org/1544.760982024-11-01T15:51:55.949586Z"Li, Ruo"https://zbmath.org/authors/?q=ai:li.ruo"Yang, Yichen"https://zbmath.org/authors/?q=ai:yang.yichen"Zhou, Yizhou"https://zbmath.org/authors/?q=ai:zhou.yizhouSummary: This work concerns with boundary conditions (BCs) of the linearized moment system for rarefied gases. Under the acoustic scaling, we analyze the boundary-layer behaviors of the moment system by resorting to a three-scale asymptotic expansion. The asymptotic analysis casts the flows into the outer solution, the viscous layer, and the Knudsen layer. Starting from the BCs of the moment system, we construct BCs for the Navier-Stokes equations by a matching requirement. The obtained BCs contain the effect of second-order terms on the velocity slip and temperature jump. For the illustrative case of the Rayleigh problem, we prove the validity of the constructed BCs through error estimates. Meanwhile, numerical tests are presented to show the performance of the constructed BCs.Velocity of viscous fingers in miscible displacement: intermediate concentrationhttps://zbmath.org/1544.761062024-11-01T15:51:55.949586Z"Bakharev, Fedor"https://zbmath.org/authors/?q=ai:bakharev.fedor-l"Enin, Aleksandr"https://zbmath.org/authors/?q=ai:enin.aleksandr"Matveenko, Sergey"https://zbmath.org/authors/?q=ai:matveenko.sergei-g"Pavlov, Dmitry"https://zbmath.org/authors/?q=ai:pavlov.dmitry-v|pavlov.dmitry-alekseevich"Petrova, Yulia"https://zbmath.org/authors/?q=ai:petrova.yulia"Rastegaev, Nikita"https://zbmath.org/authors/?q=ai:rastegaev.nikita"Tikhomirov, Sergey"https://zbmath.org/authors/?q=ai:tikhomirov.sergey-bSummary: We investigate one-phase flow in porous medium corresponding to a miscible displacement process in which the viscosity of the injected fluid is smaller than the viscosity in the reservoir fluid, which frequently leads to the formation of a mixing zone characterized by thin fingers. The mixing zone grows in time due to the difference in speed between its leading and trailing edges. The transverse flow equilibrium (TFE) model provides estimates of these speeds. We propose an enhancement for the TFE estimates, and provide its theoretical justification. It is based on the assumption that an intermediate concentration exists near the tip of the finger, which allows to reduce the integration interval in the speed estimate. Numerical simulations were conducted that corroborate the new estimates within the computational fluid dynamics model. The refined estimates offer greater accuracy than those provided by the original TFE model.Allen-Cahn-Navier-Stokes-Voigt systems with moving contact lineshttps://zbmath.org/1544.761112024-11-01T15:51:55.949586Z"Gal, Ciprian G."https://zbmath.org/authors/?q=ai:gal.ciprian-g"Grasselli, Maurizio"https://zbmath.org/authors/?q=ai:grasselli.maurizio"Poiatti, Andrea"https://zbmath.org/authors/?q=ai:poiatti.andreaSummary: We consider a diffuse interface model for an incompressible binary fluid flow. The model consists of the Navier-Stokes-Voigt equations coupled with the mass-conserving Allen-Cahn equation with Flory-Huggins potential. The resulting system is subject to generalized Navier boundary conditions for the (volume averaged) fluid velocity \(\mathbf{u}\) and to a dynamic contact line boundary condition for the order parameter \(\phi\). These boundary conditions account for the moving contact line phenomenon. We establish the existence of a global weak solution which satisfies an energy inequality. A similar result is proven for the Allen-Cahn-Navier-Stokes system. In order to obtain some higher-order regularity (w.r.t. time) we propose the Voigt approximation: in this way we are able to prove the validity of the energy identity and of the strict separation property. Thanks to this property, we can show the uniqueness of quasi-strong solutions, even in dimension three. Regularization in finite time of weak solutions is also shown.A regularized model for wetting/dewetting problems: positivity and asymptotic analysishttps://zbmath.org/1544.761242024-11-01T15:51:55.949586Z"Zhou, Zeyu"https://zbmath.org/authors/?q=ai:zhou.zeyu"Jiang, Wei"https://zbmath.org/authors/?q=ai:jiang.wei.7"Zhang, Zhen"https://zbmath.org/authors/?q=ai:zhang.zhenSummary: We consider a general regularized variational model for simulating wetting/dewetting phenomena arising from solids or fluids. The regularized model leads to the appearance of a precursor layer which covers the bare substrate, with the precursor height depending on the regularization parameter \(\varepsilon \). This model enjoys lots of advantages in analysis and simulations. With the help of the precursor layer, the spatial domain is naturally extended to a larger fixed one in the regularized model, which leads to both analytical and computational eases. There is no need to explicitly track the contact line motion, and difficulties arising from free boundary problems can be avoided. In addition, topological change events can be automatically captured. Under some mild and physically meaningful conditions, we show the positivity-preserving property of the minimizers of the regularized model. By using formal asymptotic analysis and \(\Gamma \)-limit analysis, we investigate the convergence relations between the regularized model and the classical sharp-interface model. Finally, numerical results are provided to validate our theoretical analysis, as well as the accuracy and efficiency of the regularized model.Global existence and decay rate of smooth solutions for full system of partial differential equations for three-dimensional compressible magnetohydrodynamic flowshttps://zbmath.org/1544.761292024-11-01T15:51:55.949586Z"Abdallah, Mohamed Ahmed"https://zbmath.org/authors/?q=ai:abdallah.mohamed-ahmed"Tan, Zhong"https://zbmath.org/authors/?q=ai:tan.zhong.1|tan.zhong(no abstract)Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaceshttps://zbmath.org/1544.810772024-11-01T15:51:55.949586Z"Marquette, Ian"https://zbmath.org/authors/?q=ai:marquette.ian"Zhang, Junze"https://zbmath.org/authors/?q=ai:zhang.junze"Zhang, Yao-Zhong"https://zbmath.org/authors/?q=ai:zhang.yaozhongSummary: Superintegrable systems in two-dimensional (2D) Darboux spaces were classified and it was found that there exist 12 distinct classes of superintegrable systems with quadratic integrals of motion (and quadratic symmetry algebras generated by the integrals) in the Darboux spaces. In this paper, we obtain exact solutions via purely algebraic means for the energies of all the 12 existing classes of superintegrable systems in four different 2D Darboux spaces. This is achieved by constructing the deformed oscillator realization and finite-dimensional irreducible representation of the underlying quadratic symmetry algebra generated by quadratic integrals respectively for each of the 12 superintegrable systems. We also introduce generic cubic and quintic algebras, generated respectively by linear and quadratic integrals and linear and cubic integrals, and obtain their Casimir operators and deformed oscillator realizations. As examples of applications, we present three classes of new superintegrable systems with cubic symmetry algebras in 2D Darboux spaces.Study of scalar particles through the Klein-Gordon equation under rainbow gravity effects in Bonnor-Melvin-lambda space-timehttps://zbmath.org/1544.811222024-11-01T15:51:55.949586Z"Ahmed, Faizuddin"https://zbmath.org/authors/?q=ai:ahmed.faizuddin"Bouzenada, Abdelmalek"https://zbmath.org/authors/?q=ai:bouzenada.abdelmalek(no abstract)The Poisson-Boltzmann equation. An introductionhttps://zbmath.org/1544.820022024-11-01T15:51:55.949586Z"Blossey, Ralf"https://zbmath.org/authors/?q=ai:blossey.ralfThis book provides a short but comprehensive introduction to the topic of the Poisson-Boltzmann equation. Its physical origin and properties are tightly connected with a wide variety of modern problems related to the behaviour of electrolyte systems varying from the fundamental questions of thermodynamics of systems with long-range interactions to practical issues of technology of capacitors, batteries, ionic liquid solvents, etc. The author gives an overview, which can form a clear picture of the underlying apparatus of mathematics and statistical physics.
The first chapter introduces the meanfield origin of the Poisson-Boltzmann equation considering the electrostatic potential-based representation of the Maxwell equation for a liquid medium operating with a simple electrolyte formed by a \((1:1)\) salt, NaCl for example. For this system, the derivation is provided as well as approximation at scales of spatial resolution and the potential's range, which leads to classic approaches to the theory of electrolytes. Several examples of their solutions are discussed in planar, cylindrical and spherical geometries.
The second chapter presents a more strict approach operating with the sequential line of reasoning based on the Hamiltonian for a system of the Coulomb gas of interacting ions. It leads to the partition function, which in what follows leads to the mean-field equations utilizing the saddle-point approximation for the Hamiltonian, and further to the one-loop correction. The physically specific case study includes the interface of an air-electrolyte system, which results in the possibility of taking the surface tension into consideration. For this quantity, computed results are compared with experimental ones.
The third chapter finalizes the book by refining the consideration taking into account a solvent's structure. In this case, the dielectric permittivity turns out to be nonlocal depending on the charge-density structure factor, and the Poisson-Boltzmann equation becomes integro-differential in a general case. The authors address in detail two more particular cases: point-wise and finite-size dipoles; for them, the dimensional physically interpretable picture is obtained and discussed in comparison with the mean-field approaches considered above.
Summarizing, this book can be recommended to a wide auditory of physicists as providing the ground-forming overview balanced with respect to explaining mathematical tools needed for acquiring models of ionic electrolyte systems and physical phenomena with their approximations not allowing for mathematical concepts to remain abstract.
Reviewer: Eugene Postnikov (Kursk)A DG method for the simulation of CO\(_2\) storage in saline aquiferhttps://zbmath.org/1544.860162024-11-01T15:51:55.949586Z"Riviere, Beatrice"https://zbmath.org/authors/?q=ai:riviere.beatrice-m"Yang, Xin"https://zbmath.org/authors/?q=ai:yang.xinSummary: To simulate the process of CO\(_2\) injection into deep saline aquifers, we use the isothermal two-phase two-component model, which takes mass transfer into account. We develop a new discontinuous Galerkin method called the ``partial upwind'' method for space discretization, incorporated with the backward Euler scheme for time discretization and the Newton-Raphson method for linearization. Numerical simulations show that the new method is a promising candidate for the CO\(_2\) storage problem in both homogenous and heterogenous porous media and is more robust to the standard discontinuous Galerkin method for some subsurface fluid flow problems.
For the entire collection see [Zbl 1414.00019].A consensus-based alternating direction method for mixed-integer and PDE-constrained gas transport problemshttps://zbmath.org/1544.900482024-11-01T15:51:55.949586Z"Krug, Richard"https://zbmath.org/authors/?q=ai:krug.richard"Leugering, Günter"https://zbmath.org/authors/?q=ai:leugering.gunter"Martin, Alexander"https://zbmath.org/authors/?q=ai:martin.alexander"Schmidt, Martin"https://zbmath.org/authors/?q=ai:schmidt.martin"Weninger, Dieter"https://zbmath.org/authors/?q=ai:weninger.dieterSummary: We consider dynamic gas transport optimization problems, which lead to large-scale and nonconvex mixed-integer nonlinear optimization problems (MINLPs) on graphs. Usually, the resulting instances are too challenging to be solved by state-of-the-art MINLP solvers. In this paper, we use graph decompositions to obtain multiple optimization problems on smaller blocks, which can be solved in parallel and may result in simpler classes of optimization problems because not every block necessarily contains mixed-integer or nonlinear aspects. For achieving feasibility at the interfaces of the several blocks, we employ a tailored consensus-based penalty alternating direction method. Our numerical results show that such decomposition techniques can outperform the baseline approach of just solving the overall MINLP from scratch. However, a complete answer to the question of how to decompose MINLPs on graphs in dependence of the given model is still an open topic for future research.Consensus-based optimization for saddle point problemshttps://zbmath.org/1544.902062024-11-01T15:51:55.949586Z"Huang, Hui"https://zbmath.org/authors/?q=ai:huang.hui.2"Qiu, Jinniao"https://zbmath.org/authors/?q=ai:qiu.jinniao"Riedl, Konstantin"https://zbmath.org/authors/?q=ai:riedl.konstantinSummary: In this paper, we propose consensus-based optimization for saddle point problems (CBO-SP), a novel multi-particle metaheuristic derivative-free optimization method capable of provably finding global Nash equilibria. Following the idea of swarm intelligence, the method employs two groups of interacting particles, one which performs a minimization over one variable while the other performs a maximization over the other variable. The two groups constantly exchange information through a suitably weighted average. This paradigm permits a passage to the mean-field limit, which makes the method amenable to theoretical analysis, and it allows to obtain rigorous convergence guarantees under reasonable assumptions about the initialization and the objective function, which most notably include nonconvex-nonconcave objectives. We further provide numerical evidence for the success of the algorithm.A generalized \(\Gamma\)-convergence concept for a class of equilibrium problemshttps://zbmath.org/1544.910222024-11-01T15:51:55.949586Z"Hintermüller, Michael"https://zbmath.org/authors/?q=ai:hintermuller.michael"Stengl, Steven-Marian"https://zbmath.org/authors/?q=ai:stengl.steven-marianSummary: A novel generalization of \(\Gamma\)-convergence applicable to a class of equilibrium problems is studied. After the introduction of the latter, a variety of its applications is discussed. The existence of equilibria with emphasis on Nash equilibrium problems is investigated. Subsequently, our \(\Gamma\)-convergence notion for equilibrium problems is introduced and discussed as well as applied to a class of penalized generalized Nash equilibrium problems and quasi-variational inequalities. The work ends with a comparison of our results to previous generalizations in the literature.On Lipschitz solutions of mean field games master equationshttps://zbmath.org/1544.910432024-11-01T15:51:55.949586Z"Bertucci, Charles"https://zbmath.org/authors/?q=ai:bertucci.charles"Lasry, Jean-Michel"https://zbmath.org/authors/?q=ai:lasry.jean-michel"Lions, Pierre-Louis"https://zbmath.org/authors/?q=ai:lions.pierre-louisSummary: We develop a theory of existence and uniqueness of solutions of MFG master equations when the initial condition is Lipschitz continuous. Namely, we show that as long as the solution of the master equation is Lipschitz continuous in space, it is uniquely defined. Because we do not impose any structural assumptions, such as monotonicity for instance, there is a maximal time of existence for the notion of solution we provide. We analyze three cases: the case of a finite state space, the case of master equation set on a Hilbert space, and finally on the set of probability measures, all in cases involving common noises. In the last case, the Lipschitz continuity we refer to is on the gradient of the value function with respect to the state variable of the player.Discounted mean-field game model of a dense static crowd with variable information crossed by an intruderhttps://zbmath.org/1544.910442024-11-01T15:51:55.949586Z"Butano, Matteo"https://zbmath.org/authors/?q=ai:butano.matteo"Appert-Rolland, Cécile"https://zbmath.org/authors/?q=ai:appert-rolland.cecile"Ullmo, Denis"https://zbmath.org/authors/?q=ai:ullmo.denisSummary: It was demonstrated in [1] that the anticipation pattern displayed by a dense crowd crossed by an intruder can be successfully described by a minimal mean-field games model. However, experiments show that changes in the pedestrian knowledge significantly modify the dynamics of the crowd. Here, we show that the addition of a single parameter, the discount factor \(\gamma\), which gives a lower weight to events distant in time, is sufficient to observe the whole variety of behaviors observed in the experiments. We present a comparison between the discounted MFG and the experimental data, also providing new analytic results and insight about how the introduction of \(\gamma\) modifies the model.Learning, mean field approximations, and phase transitions in auction modelshttps://zbmath.org/1544.911572024-11-01T15:51:55.949586Z"Pinasco, Juan Pablo"https://zbmath.org/authors/?q=ai:pinasco.juan-pablo"Saintier, Nicolas"https://zbmath.org/authors/?q=ai:saintier.nicolas"Kind, Martin"https://zbmath.org/authors/?q=ai:kind.martinIn this article, the authors study an agent-based model for multi-round, pay as bid, sealed bid reverse auctions using techniques from partial differential equations and statistical mechanics tools under the assumption that in each round a fixed fraction of bidders is awarded, and bidders learn from round to round using simple microscopic rules, adjusting myopically their bid according to their performance.
The problem of an unadvertised agreement between buyers of non-exclusive rights at a spectrum auction for the agreed formation of price bids is considered from the viewpoint of game theory and operations research. The simulations show that bidders coordinate in the sense that they tend to bid the same value in the long-time limit. In Section 6 the authors analyze the case of a heterogeneous cost distribution among the bidders showing that the phase transition at \(\rho = 2\) persists in this more general setting. When \(\rho = 2\), an exact formula for the final mean price involving the cost distribution is obtained heuristically building on the intuition developed so far, and which agrees very well with the simulations.
Reviewer: Nikolay Kyurkchiev (Plovdiv)Nonparametric measurement of productivity growth and technical changehttps://zbmath.org/1544.911702024-11-01T15:51:55.949586Z"Ray, Subhash C."https://zbmath.org/authors/?q=ai:ray.subhash-cSummary: The three main components of productivity change are technical change, change in technical efficiency, and returns to scale effects. Solow measured productivity change at the macroeconomic level as the difference between the growth rates of output and input which, under constant returns to scale and in the absence of any technical inefficiency, is a measure of technical change. The focus in this monograph is on the individual firm and both technical inefficiency and variable returns to scale are accommodated.
In neoclassical production economics, productivity change can be measured alternatively from the production, cost, profit, or distance functions. In continuous time analysis, one measures the rates of productivity and technical change. In discrete time, one measures \textit{indexes} of productivity and technical change over time. This work describes the Tornqvist, Fisher, and Malmquist productivity indexes along with the Luenberger productivity indicator and a geometric Young index and how they relate to one another. The relevant nonparametric DEA models for measuring the different productivity indexes are formulated for nonparametric analysis of productivity, technical change, and change in efficiency.Competition and market dynamics in duopoly: the effect of switching costshttps://zbmath.org/1544.912082024-11-01T15:51:55.949586Z"Yang, Yang"https://zbmath.org/authors/?q=ai:yang.yang.19|yang.yang.70|yang.yang.10|yang.yang.22|yang.yang.44|yang.yang.42|yang.yang.16|yang.yang.41|yang.yang|yang.yang.23|yang.yang.9|yang.yang.54|yang.yang.49|yang.yang.43|yang.yang.50|yang.yang.51|yang.yang.29|yang.yang.7|yang.yang.45|yang.yang.40|yang.yang.53|yang.yang.6|yang.yang.14|yang.yang.48|yang.yang.20"Wu, Cheng-Hung"https://zbmath.org/authors/?q=ai:wu.cheng-hungSummary: A dynamic game framework is developed to study market dynamics between two manufacturers/service providers competing on pricing and switching costs. In this game, a portion of consumers may choose to upgrade their products by repurchasing from one of the providers in each period. The switching cost is the one-time costs when consumers ``switch'' from one provider to another. Switching costs provide consumers an incentive to continue buying from the same firm even if its competitors offer functionally identical but incompatible products. In practice, the switching costs can be increased or decreased by firms through designing products. A mixed logit demand model, which can arbitrarily closely approximate any discrete choice behavior of consumers, is adopted to characterize the dynamic market evolution under stochastically varying consumer preferences. We find that switching costs are usually beneficial to the firm with a dominant market share. Moreover, large switching costs can be detrimental to the firm with a disadvantaged market share, so it wants to decrease switching costs. On the contrary, small switching costs have a negative effect on the demand of the firm with a weak market share but benefit its profit by leading a high price. We implement a simulation study to validate our theoretical results on market dynamics.Pricing of timer digital power options based on stochstic volatilityhttps://zbmath.org/1544.913292024-11-01T15:51:55.949586Z"Ha, Mijin"https://zbmath.org/authors/?q=ai:ha.mijin"Park, Sangmin"https://zbmath.org/authors/?q=ai:park.sangmin"Kim, Donghyun"https://zbmath.org/authors/?q=ai:kim.donghyun.2|kim.donghyun.3|kim.donghyun|kim.donghyun.1"Yoon, Ji-Hun"https://zbmath.org/authors/?q=ai:yoon.ji-hunSummary: Timer options are financial instruments proposed by Société Générale Corporate and Investment Banking in 2007. Unlike vanilla options, where the expiry date is fixed, the expiry date of timer options is determined by the investor's choice, which is in linked to a variance budget. In this study, we derive a pricing formula for hybrid options that combine timer options, digital options, and power options, considering an environment where volatility of an underlying asset follows a fast-mean-reverting process. Additionally, we aim to validate the pricing accuracy of these analytical formulas by comparing them with the results obtained from Monte Carlo simulations. Finally, we conduct numerical studies on these options to analyze the impact of stochastic volatility on option's price with respect to various model parameters.How good is Black-Scholes-Merton, really?https://zbmath.org/1544.913392024-11-01T15:51:55.949586Z"Wilmott, P."https://zbmath.org/authors/?q=ai:wilmott.paulSummary: I consider the Black-Scholes-Merton option-pricing model from several angles, including personal, technical and, most importantly, from the perspective of a paradigm-shifting mathematical formula.
For the entire collection see [Zbl 1543.91001].On the convergence of a Crank-Nicolson fitted finite volume method for pricing American bond optionshttps://zbmath.org/1544.913582024-11-01T15:51:55.949586Z"Gan, Xiaoting"https://zbmath.org/authors/?q=ai:gan.xiaoting"Xu, Dengguo"https://zbmath.org/authors/?q=ai:xu.dengguoSummary: This paper develops and analyses a Crank-Nicolson fitted finite volume method to price American options on a zero-coupon bond under the Cox-Ingersoll-Ross (CIR) model governed by a partial differential complementarity problem (PDCP). Based on a penalty approach, the PDCP results in a nonlinear partial differential equation (PDE). We then apply a fitted finite volume method for the spatial discretization along with a Crank-Nicolson time-stepping scheme for the PDE, which results in a nonlinear algebraic equation. We show that this scheme is consistent, stable, and monotone, and hence, the convergence of the numerical solution to the viscosity solution of the continuous problem is guaranteed. To solve the system of nonlinear equations effectively, an iterative algorithm is established and its convergence is proved. Numerical experiments are presented to demonstrate the accuracy, efficiency, and robustness of the new numerical method.Chemotactic cell aggregation viewed as instability and phase separationhttps://zbmath.org/1544.920292024-11-01T15:51:55.949586Z"Choi, Kyunghan"https://zbmath.org/authors/?q=ai:choi.kyunghan"Kim, Yong-Jung"https://zbmath.org/authors/?q=ai:kim.yongjungSummary: The paper focuses on the pattern formation of a chemotactic cell aggregation model with a mechanism that density suppresses motility. The model exhibits four types of cell aggregation patterns: single-point peaks, hot spots, cold spots, and stripes, depending on the parameters and mean density. The analysis is performed in two ways. First, traditional instability analysis reveals the existence of two critical densities. This local analysis shows patterns emerge if the initial mean density lies between the two values. Second, a phase separation method using van der Waals' double well potential reveals that pattern formation is possible in a bigger parameter regime that includes the one identified by the local analysis. This non-local analysis shows that pattern formation occurs beyond the parameter regimes of the classical local instability analysis.A non-local kinetic model for cell migration: a study of the interplay between contact guidance and steric hindrancehttps://zbmath.org/1544.920302024-11-01T15:51:55.949586Z"Conte, Martina"https://zbmath.org/authors/?q=ai:conte.martina"Loy, Nadia"https://zbmath.org/authors/?q=ai:loy.nadiaSummary: We propose a non-local model for contact guidance and steric hindrance depending on a single external cue, namely the extracellular matrix, that affects in a twofold way the polarization and speed of motion of the cells. We start from a microscopic description of the stochastic processes underlying the cell re-orientation mechanism related to the change of cell speed and direction. Then, we formally derive the corresponding kinetic model that implements exactly the prescribed microscopic dynamics, and, from it, it is possible to deduce the macroscopic limit in the appropriate regime. Moreover, we test our model in several scenarios. In particular, we numerically investigate the minimal microscopic mechanisms that are necessary to reproduce cell dynamics by comparing the outcomes of our model with some experimental results related to breast cancer cell migration. This allows us to validate the proposed modeling approach and to highlight its capability of predicting qualitative cell behaviors in diverse heterogeneous microenvironments.A biased random walk approach for modeling the collective chemotaxis of neural crest cellshttps://zbmath.org/1544.920312024-11-01T15:51:55.949586Z"Freingruber, Viktoria"https://zbmath.org/authors/?q=ai:freingruber.viktoria"Painter, Kevin J."https://zbmath.org/authors/?q=ai:painter.kevin-j"Ptashnyk, Mariya"https://zbmath.org/authors/?q=ai:ptashnyk.mariya"Schumacher, Linus J."https://zbmath.org/authors/?q=ai:schumacher.linus-jA model for the collective motion of neural crest cells is designed, based on the description of the cell motion by a biased random walk in discrete space and time. The bias is mediated by internal state variables which evolve continuously in time and spatial anisotropy is taken into account by assuming that each cell is characterized by its center \(P\) located on a two-dimensional discrete lattice with mesh size \(h>0\) and its ``membrane'' \(\{P\pm(h,0),P\pm(0,h)\}\) consisting of its nearest neighbors in the lattice. In particular, the motion of the cell is enhanced in the direction of higher concentrations of the internal state variable. The latter is subject to extracellular mechanisms, including the influence of a time-independent distribution of a chemoattractant, an increase due to co-attraction between cells, and a decrease due to contact inhibition of motion, spatial constraints, and degradation. The resulting model provides the position \(P^i(t_n)\) of the \(i\)-th cell (\(1\le i \le N\)) at time \(t_n\), \(n\ge 1\), along with the time evolution of the internal variable state \(C^i(\mathbf{x},t)\) in the \(i\)-th cell for \(t\in [t_n,t_{n+1})\) and \(\mathbf{x}\in \{ P^i(t_n), P^i(t_n)\pm (h,0), P^i(t_n)\pm (0,h)\}\). Several numerical simulations are performed and discussed, with some emphasis on the role of the chemoattractant. Under suitable assumptions, an approximating PDE system is also derived.
Reviewer: Philippe Laurençot (Chambéry)Ion-concentration gradients induced by synaptic input increase the voltage depolarization in dendritic spineshttps://zbmath.org/1544.920352024-11-01T15:51:55.949586Z"Eberhardt, Florian"https://zbmath.org/authors/?q=ai:eberhardt.florianSummary: The vast majority of excitatory synaptic connections occur on dendritic spines. Due to their extremely small volume and spatial segregation from the dendrite, even moderate synaptic currents can significantly alter ionic concentrations. This results in chemical potential gradients between the dendrite and the spine head, leading to measurable electrical currents. In modeling electric signals in spines, different formalisms were previously used. While the cable equation is fundamental for understanding the electrical potential along dendrites, it only considers electrical currents as a result of gradients in electrical potential. The Poisson-Nernst-Planck (PNP) equations offer a more accurate description for spines by incorporating both electrical and chemical potential. However, solving PNP equations is computationally complex. In this work, diffusion currents are incorporated into the cable equation, leveraging an analogy between chemical and electrical potential. For simulating electric signals based on this extension of the cable equation, a straightforward numerical solver is introduced. The study demonstrates that this set of equations can be accurately solved using an explicit finite difference scheme. Through numerical simulations, this study unveils a previously unrecognized mechanism involving diffusion currents that amplify electric signals in spines. This discovery holds crucial implications for both numerical simulations and experimental studies focused on spine neck resistance and calcium signaling in dendritic spines.A model of gastric mucosal pH regulation: extending sensitivity analysis using Sobol' indices to understand higher momentshttps://zbmath.org/1544.920402024-11-01T15:51:55.949586Z"Aggarwal, Manu"https://zbmath.org/authors/?q=ai:aggarwal.manu"Lewis, Owen"https://zbmath.org/authors/?q=ai:lewis.owen-l"Jarrett, Angela"https://zbmath.org/authors/?q=ai:jarrett.angela-m"Hussaini, M. Y."https://zbmath.org/authors/?q=ai:hussaini.m-y"Cogan, N. G."https://zbmath.org/authors/?q=ai:cogan.nicholas-gSummary: Several recent theoretical studies have indicated that a relatively simple secretion control mechanism in the epithelial cells lining the stomach may be responsible for maintaining a neutral (healthy) pH adjacent to the stomach wall, even in the face of enormous electrodiffusive acid transport from the interior of the stomach. Subsequent work used Sobol' indices (SIs) to quantify the degree to which this secretion mechanism is ``self-regulating'' i.e. the degree to which the wall pH is held neutral as mathematical parameters vary. However, questions remain regarding the nature of the control that specific parameters exert over the maintenance of a healthy stomach wall pH. Studying the sensitivity of higher moments of the statistical distribution of a model output can provide useful information, for example, how one parameter may skew the distribution towards or away from a physiologically advantageous regime. In this work, we prove a relationship between SIs and the higher moments and show how it can potentially reduce the cost of computing sensitivity of said moments. We define \(\gamma\)-indices to quantify sensitivity of variance, skewness, and kurtosis to the choice of value of a parameter, and we propose an efficient strategy that uses both SIs and \(\gamma\)-indices for a more comprehensive sensitivity analysis. Our analysis uncovers a control parameter which governs the ``tightness of control'' that the secretion mechanism exerts on wall pH. Finally, we discuss how uncertainty in this parameter can be reduced using expert information about higher moments, and speculate about the physiological advantage conferred by this control mechanism.A local continuum model of cell-cell adhesionhttps://zbmath.org/1544.920502024-11-01T15:51:55.949586Z"Falcó, C."https://zbmath.org/authors/?q=ai:falco.carles"Baker, R. E."https://zbmath.org/authors/?q=ai:baker.ruth-e"Carrillo, J. A."https://zbmath.org/authors/?q=ai:carrillo.jose-antonioSummary: Cell-cell adhesion is one the most fundamental mechanisms regulating collective cell migration during tissue development, homeostasis, and repair, allowing cell populations to self-organize and eventually form and maintain complex tissue shapes. Cells interact with each other via the formation of protrusions or filopodia and they adhere to other cells through binding of cell surface proteins. The resulting adhesive forces are then related to cell size and shape and, often, continuum models represent them by nonlocal attractive interactions. In this paper, we present a new continuum model of cell-cell adhesion which can be derived from a general nonlocal model in the limit of short-range interactions. This new model is local, resembling a system of thin-film type equations, with the various model parameters playing the role of surface tensions between different cell populations. Numerical simulations in one and two dimensions reveal that the local model maintains the diversity of cell sorting patterns observed both in experiments and in previously used nonlocal models. In addition, it also has the advantage of having explicit stationary solutions, which provides a direct link between the model parameters and the differential adhesion hypothesis.Soliton excitations in a twist-opening nonlinear DNA modelhttps://zbmath.org/1544.920542024-11-01T15:51:55.949586Z"Bugay, Alexander"https://zbmath.org/authors/?q=ai:bugay.aleksandr-nSummary: Most of nonlinear DNA models introduce only one degree of freedom, depending on the process one aims at modeling. For example, models for DNA denaturation, e.g., well-known Peyrard-Bishop model, consider a radial degree of freedom. Yakushevich model and its derivations dealing with the DNA structure modification met in the transcription process consider angular degrees of freedom. In contrast to these wide classes of DNA models, the so-called twist-opening model takes into account two degrees of freedom: radial stretching of hydrogen bonds between the base pairs and angular twist of double helix. Such an approach enables much better reproduction of geometrical constraints in a molecule and a variety of dynamical processes like melting and transcription in a single model. Here, we will analyze various types of nonlinear solitary waves emerging in a twist-opening DNA model. We will show that this type of model, in particular cases of dynamical behavior, can be reduced to well-known integrable nonlinear wave equations: nonlinear Schrödinger equation and Korteweg-de Vries equation. Different types of solitons will be obtained including oscillating ``bright'' and ``dark'' envelope solitons as well as bell-shaped solitons.
For the entire collection see [Zbl 1520.92003].Nonlinear dynamics of DNA chain with long-range interactionshttps://zbmath.org/1544.920572024-11-01T15:51:55.949586Z"Okaly, Joseph Brizar"https://zbmath.org/authors/?q=ai:okaly.joseph-brizar"Nkomom, Théodule Nkoa"https://zbmath.org/authors/?q=ai:nkomom.theodule-nkoaSummary: The present chapter is devoted to the investigation of the nonlinear dynamics of DNA molecules in the helicoidal Peyrard-Bishop DNA model with long-range interactions. The power law long-range interactions with distance dependence \(|l|^{-s}\) on the elastic coupling constant between different DNA base pair units are considered. We take into account the Stokes and the hydrodynamic damping forces in the long-range approximation. In the short wavelength modes, using the discrete difference operator technique, we show that the discrete lattice equation of motion is reduced to the complex Ginzburg-Landau equation, allowing breather soliton solutions. In the non-viscous limit, the system is shown to be governed by the nonlinear Schrödinger equation. We considered the relevant case \(s=3\), and showed analytically, that the method leads to a more appropriate expression for the breather soliton parameters as compared to the popular semi-discrete approximation.
For the entire collection see [Zbl 1520.92003].Mathematical models of acoustically induced vaporization of encapsulated nanodropletshttps://zbmath.org/1544.920792024-11-01T15:51:55.949586Z"Jiang, K."https://zbmath.org/authors/?q=ai:jiang.kaifeng|jiang.kaiyu|jiang.keyu|jiang.kai|jiang.kebei|jiang.kyle|jiang.ke|jiang.kang|jiang.keqin|jiang.kun|jiang.kaizhong|jiang.kaige|jiang.kerui|jiang.kebin|jiang.kui|jiang.keni|jiang.kaiqiang|jiang.katie|jiang.keyuan|jiang.kaijie|jiang.keshen|jiang.kaiwen|jiang.kangkang|jiang.kaichun|jiang.kaichen|jiang.kechun|jiang.kangming|jiang.kan|jiang.kexia"Ghasemi, M."https://zbmath.org/authors/?q=ai:ghasemi.mahdieh|ghasemi.mojtaba|ghasemi.mohammadreza|ghasemi.mohammad-reza.1|ghasemi.mohsen|ghasemi.mohammad-mahdi|ghasemi.masood|ghasemi.mina|ghasemi.mahnaz|ghasemi.mostafa|ghasemi.mohammad-reza|ghasemi.mahsa|ghasemi.maryam|ghasemi.mehdi|ghasemi.mohamad|ghasemi.mohammad-hassan|ghasemi.mehdi.1|labbaf-ghasemi.mohammad-hussein|ghasemi.mohammad-s"Yu, A."https://zbmath.org/authors/?q=ai:yu.anbin|yu.aimei|yu.ami|yu.anyu|yu.aolin|yu.anqi|yu.aiai|yu.aimin|yu.anxi|yu.aihui|yu.angning|yu.aibing|yu.ada|yu.xin|yu.aijun|yu.alexandria|yu.alexander|yu.anchi|yu.alan|yu.anlan|yu.alex|yu.anxu|yu.aobo|yu.aihua|yu.aiwen|yu.annan|yu.aiming|yu.ankang|yu.anying"Sivaloganathan, S."https://zbmath.org/authors/?q=ai:sivaloganathan.siv|sivaloganathan.sivabalSummary: The induced vaporization of nanodroplets has shown promise to be used in various biomedical applications such as contrast imaging, embolotherapy, as well as the enabling of targeted drug delivery in chemotherapy. This process works through the introduction of encapsulated nanodroplets into the bloodstream. These nanodroplets contain an appropriately chosen liquid that is metastable at human body temperature but can be induced to vaporize through externally applied ultrasound. This paper presents the mathematical modelling of the ADV process using the Rayleigh-Plesset framework that has been generalized to incorporate the effects of viscosity, surface tension, heat transfer, etc. Finally, it presents two models of the encapsulating shell which are then incorporated into the framework to more accurately model the behaviour of the encapsulated nanodroplet within an acoustic field.
For the entire collection see [Zbl 1515.92004].Computational modeling of cancer response to oncolytic virotherapy: improving the effectiveness of viral spread and anti tumor efficacyhttps://zbmath.org/1544.920812024-11-01T15:51:55.949586Z"Lefraich, H."https://zbmath.org/authors/?q=ai:lefraich.hamidSummary: Oncolytic viruses (OV) are genetically engineered viruses that can selectively infect cancer cells, multiply inside them, destroy them and spread to further tumor cells without causing harm to normal healthy cells. They could be used as a therapy approach for cancer treatment that is promising in principle, however the success of oncolytic virotherapy is dampened by the presence of the extracellular matrix (ECM). In fact, the ECM has been recognized as a major barrier for anti-tumor efficacy as it plays a pivotal role in inhibiting virus spread. In this work, we develop a new mathematical model that can improve understanding of the role of viral diffusivity to provide insights that can be useful for the further development of this therapy approach. The spatio-temporal model is formulated in terms of equations that take into account the interaction between uninfected cancer cells, infected cancer cells, extracellular matrix (ECM) and oncolytic virus. Finally, numerical investigations were carried out for different scenarios. For the considered parameter regimes, the numerical simulations show that the viral therapy leads to control and decrease of the overall tumor expansion.
For the entire collection see [Zbl 1515.92004].A computational framework for the administration of 5-aminolevulinic acid before glioblastoma surgeryhttps://zbmath.org/1544.920892024-11-01T15:51:55.949586Z"Zeng, Jia"https://zbmath.org/authors/?q=ai:zeng.jia"Moore, Nicholas J."https://zbmath.org/authors/?q=ai:moore.nicholas-jSummary: 5-Aminolevulinic acid (5-ALA) is the only fluorophore approved by the FDA as an intraoperative optical imaging agent for fluorescence-guided surgery in patients with glioblastoma. The dosing regimen is based on rodent tests where a maximum signal occurs around 6\,h after drug administration. Here, we construct a computational framework to simulate the transport of 5-ALA through the stomach, blood, and brain, and the subsequent conversion to the fluorescent agent protoporphyrin IX at the tumor site. The framework combines compartmental models with spatially-resolved partial differential equations, enabling one to address questions regarding quantity and timing of 5-ALA administration before surgery. Numerical tests in two spatial dimensions indicate that, for tumors exceeding the detection threshold, the time to peak fluorescent concentration is 2-7\,h, broadly consistent with the current surgical guidelines. Moreover, the framework enables one to examine the specific effects of tumor size and location on the required dose and timing of 5-ALA administration before glioblastoma surgery.Existence of closed trajectories in Lotka-Volterra systems in \(\mathbb{R}^+_n\)https://zbmath.org/1544.921242024-11-01T15:51:55.949586Z"Bratus, A."https://zbmath.org/authors/?q=ai:bratus.a-v|bratus.alexander-s"Tikhomirov, V."https://zbmath.org/authors/?q=ai:tikhomirov.v-n|tikhomirov.vladimir-m|tikhomirov.v-v|tikhomirov.v-p"Isaev, R."https://zbmath.org/authors/?q=ai:isaev.rFor the entire collection see [Zbl 1515.92004].Mathematical models of predators and prey with lateralityhttps://zbmath.org/1544.921492024-11-01T15:51:55.949586Z"Nakajima, Mifuyu"https://zbmath.org/authors/?q=ai:nakajima.mifuyuSummary: Laterality in fish is considered one of the best examples of frequency-dependent selection. In scale-eating fish, the most abundant morphological type (i.e., right or left type) of a species consumes less food because prey fish tend to defend themselves more against the dominant type; therefore, the rarer type of scale eater can consume more, becoming the dominant type. This idea is well supported by one mathematical model that illustrates changes in the fitness of each type of scale eater due to prey defense. Another model explains frequency-dependent selection between the two laterality types in both piscivorous fish and their prey. This model describes the population dynamics of each laterality type in predators and prey that exhibit biased predation, i.e., the right-type predator mainly consumes the left-type prey, and vice versa. The laterality ratio of each species obtained from this model oscillates periodically, showing continuous alternations of fitness between the two laterality types. In addition, the model suggests that monomorphism is not sustained in food webs with omnivory. Therefore, these models indicate that biased predation between different laterality types may maintain dimorphism.
For the entire collection see [Zbl 1515.92003].Mathematical modeling and numerical analysis of HIV-1 infection with long-lived infected cells during combination therapy and humoral immunityhttps://zbmath.org/1544.921842024-11-01T15:51:55.949586Z"Hajhouji, Zakaria"https://zbmath.org/authors/?q=ai:hajhouji.zakaria"El Younoussi, Majda"https://zbmath.org/authors/?q=ai:el-younoussi.majda"Hattaf, Khalid"https://zbmath.org/authors/?q=ai:hattaf.khalid"Yousfi, Noura"https://zbmath.org/authors/?q=ai:yousfi.nouraSummary: In this work, we develop a mathematical model formulated by partial differential equations (PDEs) describing the HIV-1 dynamics with four discrete delays, general incidence rate, humoral immune response in the presence of therapy, and two types of infected cells. A numerical method is proposed to discretize the developed PDE model. The mathematical and numerical analysis of the HIV-1 infection model is rigorously investigated. Finally, numerical simulations are presented to illustrate our results.
For the entire collection see [Zbl 1531.92006].Dynamics of an SIS epidemic model with no vertical transmissionhttps://zbmath.org/1544.921882024-11-01T15:51:55.949586Z"Kovács, Sándor"https://zbmath.org/authors/?q=ai:kovacs.sandor"György, Szilvia"https://zbmath.org/authors/?q=ai:gyorgy.szilvia"Gyúró, Noémi"https://zbmath.org/authors/?q=ai:gyuro.noemiSummary: We study a population model for an infectious disease that spreads in the host population through standard incidence and no vertical transmission. We show that the global dynamics are completely determined by the basic reproduction number \(\mathscr{R}_0\). If \(\mathscr{R}_0 < 1\), the disease-free equilibrium is globally asymptotically stable, and the disease always dies out. If \(\mathscr{R}_0 > 1\), a new (endemic) equilibrium emerges. Sensitivity analysis is performed on this epidemic threshold value, and it is then used to show that at its critical value bifurcation takes place. This bifurcation is forward: a super-threshold endemic equilibrium exists, the global asymptotic stability of which is also shown. Improving discretization (nonstandard finite difference scheme), our results are corresponding to those in the original continuous model.
For the entire collection see [Zbl 1531.92006].Global dynamics of a periodic viral infection model incorporating diffusion and antiretroviral therapyhttps://zbmath.org/1544.922092024-11-01T15:51:55.949586Z"Xu, Jinhu"https://zbmath.org/authors/?q=ai:xu.jinhuSummary: A new viral infection model with cell-to-cell infection and periodic antiretroviral therapy was proposed and analyzed. With the help of defined basic reproduction number \(\mathfrak{R}_0\), we proved that if \(\mathfrak{R}_0<1\), then the infection-free steady state is globally attractive. If \(\mathfrak{R}_0>1\), then the infection is uniformly persistent. Furthermore, the globally asymptotical stability of infection-free steady state has been established for the critical case of \(\mathfrak{R}_0=1\). Also, the global asymptotic stability of the infection steady state for a special case of the model has been established by applying the method of Lyapunov. Numerical simulations show that increasing the drug efficacy for blocking virus-to-cell infection, cell-to-cell infection and producing non-infectious virus contributes to weakening the severity of viral infection.Dynamic exploration and control of bifurcation in a fractional-order Lengyel-Epstein model owing time delayhttps://zbmath.org/1544.922442024-11-01T15:51:55.949586Z"Li, Peiluan"https://zbmath.org/authors/?q=ai:li.peiluan"Lu, Yuejing"https://zbmath.org/authors/?q=ai:lu.yuejing"Xu, Changjin"https://zbmath.org/authors/?q=ai:xu.changjin"Ren, Jing"https://zbmath.org/authors/?q=ai:ren.jingSummary: Delayed differential equation plays a vital role in revealing the dynamics of chemical reaction law. In this work, we propose a novel fractional-order Lengyel-Epstein model owing time delay. By regarding the delay as parameter and investigating the distribution of roots of the associated characteristic equation of the formulated fractional-order delayed Lengyel-Epstein model, we set up a new delay-dependent criterion on stability and bifurcation of the involved fractional-order delayed Lengyel-Epstein model. Making use of nonlinear delayed feedback controller, we can effectually control the stability domain and the time of bifurcation phenomenon of the formulated fractional-order delayed Lengyel-Epstein model. Taking advantage of hybrid controller, we are able to adjust the stability domain and the time of bifurcation phenomenon of the established fractional-order delayed Lengyel-Epstein model. The study shows that delay is a vital factor which affects the stability and bifurcation behavior of the addressed fractional-order delayed Lengyel-Epstein model. In order to illustrate the rationality of the acquired theoretical outcomes, we execute Matlab simulations to check this fact. The gained outcomes in this work are absolutely innovative and possess enormous theoretical significance in adjusting concentrations of different chemical substance.The null controllability of transmission wave-Schrödinger system with a boundary controlhttps://zbmath.org/1544.930512024-11-01T15:51:55.949586Z"Guo, Ya-Ping"https://zbmath.org/authors/?q=ai:guo.yaping"Wang, Jun-Min"https://zbmath.org/authors/?q=ai:wang.junmin"Wang, Jing"https://zbmath.org/authors/?q=ai:wang.jing.32"Zhao, Dong-Xia"https://zbmath.org/authors/?q=ai:zhao.dongxiaSummary: This paper is devoted to investigate the null controllability of the transmission wave-Schrödinger system with only one boundary control. The domain of the system consists of two bounded intervals, where the wave and Schrödinger equations evolve, respectively. Two kinds of transmission conditions are considered: one is the simple continuous transmission, and by using the HUM method, the null controllability of the system is derived in the Hilbert space when a boundary control is added only on the wave equation. The other case is that the Schrödinger state is associated with the velocity of the wave, and we establish the null controllability of the system. It is found that the second space has more regularity.Small-time global approximate controllability for incompressible MHD with coupled Navier slip boundary conditionshttps://zbmath.org/1544.930592024-11-01T15:51:55.949586Z"Rissel, Manuel"https://zbmath.org/authors/?q=ai:rissel.manuel"Wang, Ya-Guang"https://zbmath.org/authors/?q=ai:wang.yaguang|wang.ya-guangSummary: We study the small-time global approximate controllability for incompressible magnetohydrodynamic (MHD) flows in smoothly bounded two- or three-dimensional domains. The controls act on arbitrary nonempty open portions of each connected boundary component, while linearly coupled Navier slip-with-friction conditions are imposed along the uncontrolled parts of the boundary. Some choices for the friction coefficients give rise to interacting velocity and magnetic field boundary layers. We obtain sufficient dissipation properties of these layers by a detailed analysis of the corresponding asymptotic expansions. For certain friction coefficients, or if the obtained controls are not compatible with the induction equation, an additional pressure-like term appears. We show that such a term does not exist for problems defined in planar simply-connected domains and various choices of Navier slip-with-friction boundary conditions.Stokes-Dirac structures for distributed parameter port-Hamiltonian systems: an analytical viewpointhttps://zbmath.org/1544.933032024-11-01T15:51:55.949586Z"Brugnoli, Andrea"https://zbmath.org/authors/?q=ai:brugnoli.andrea"Haine, Ghislain"https://zbmath.org/authors/?q=ai:haine.ghislain"Matignon, Denis"https://zbmath.org/authors/?q=ai:matignon.denisSummary: In this paper, we prove that a large class of linear evolution partial differential equations defines a Stokes-Dirac structure over Hilbert spaces. To do so, the theory of boundary control system is employed. This definition encompasses problems from mechanics that cannot be handled by the geometric setting given in the seminal paper by
\textit{A. J. van der Schaft} and \textit{B. M. Maschke} [J. Geom. Phys. 42, No. 1--2, 166--194 (2002; Zbl 1012.70019)].
Many worked-out examples stemming from continuum mechanics and physics are presented in detail, and a particular focus is given to the functional spaces in duality at the boundary of the geometrical domain. For each example, the connection between the differential operators and the associated Hilbert complexes is illustrated.Disturbance rejection approaches of Korteweg-de Vries-Burgers equation under event-triggering mechanismhttps://zbmath.org/1544.933062024-11-01T15:51:55.949586Z"Kang, Wen"https://zbmath.org/authors/?q=ai:kang.wen"Zhang, Jing"https://zbmath.org/authors/?q=ai:zhang.jing.233"Wang, Jun-Min"https://zbmath.org/authors/?q=ai:wang.junminSummary: In this paper, disturbance rejection approaches are suggested to stabilize Korteweg-de Vries-Burgers (KdVB) equation under the averaged measurements. Here two approaches -- active disturbance rejection control (ADRC) and disturbance observer-based control (DOBC), are introduced to reject the external unknown disturbance actively. The main challenging issue is to design the effective extended state observer (ESO)/disturbance observer (DO) for KdVB equation respectively. As for ADRC strategy, the disturbance is rejected on the basis of an ESO. As for DOBC strategy, a DO is constructed to estimate the disturbance formulated by an exogenous system. Continuous-time and event-triggered anti-disturbance controllers are further presented for distributed stabilization of KdVB system. To significantly reduce the amount of control updates, an event-triggering mechanism is utilized and the Zeno behaviour is avoided. Sufficient stability conditions are established via Lyapunov functional method. The effectiveness of proposed approaches is verified by simulation results.On the boundary stabilization of the KdV-KdV system with time-dependent delayhttps://zbmath.org/1544.935962024-11-01T15:51:55.949586Z"Capistrano-Filho, Roberto de A."https://zbmath.org/authors/?q=ai:capistrano-filho.roberto-de-a"Chentouf, Boumediène"https://zbmath.org/authors/?q=ai:chentouf.boumediene"Martinez, Victor H. Gonzalez"https://zbmath.org/authors/?q=ai:martinez.victor-h-gonzalez"Muñoz, Juan Ricardo"https://zbmath.org/authors/?q=ai:munoz.juan-ricardo.1Summary: The boundary stabilization problem of the Boussinesq KdV-KdV type system is investigated in this paper. An appropriate boundary feedback law consisting of a linear combination of a damping mechanism and a delay term is designed. Then, considering time-varying delay feedback together with a smallness restriction on the length of the spatial domain and the initial data, we show that the problem under consideration is well-posed. The proof combines Kato's approach and the fixed-point argument. Last but not least, we prove that the energy of the linearized KdV-KdV system decays exponentially by employing the Lyapunov method.Stabilization of linear KdV equation with boundary time-delay feedback and internal saturationhttps://zbmath.org/1544.936692024-11-01T15:51:55.949586Z"Taboye, Ahmat"https://zbmath.org/authors/?q=ai:taboye.ahmat-mahamat"Ennouari, Toufik"https://zbmath.org/authors/?q=ai:ennouari.toufikSummary: This research studies the stabilization of the linear KdV equation with time-delay on boundary feedback in the presence of a saturated source term. Under certain hypotheses, the proof of well-posedness is established. The result of exponential stability is demonstrated using an appropriate Lyapunov functional.Optimal stopping of conditional McKean-Vlasov jump diffusionshttps://zbmath.org/1544.937612024-11-01T15:51:55.949586Z"Agram, Nacira"https://zbmath.org/authors/?q=ai:agram.nacira"Øksendal, Bernt"https://zbmath.org/authors/?q=ai:oksendal.bernt-karstenSummary: The purpose of this paper is to study the optimal stopping problem of conditional McKean-Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean-Vlasov jump diffusions, for short). We obtain sufficient variational inequalities for a function to be the value function of such a problem and for a stopping time to be optimal.
The key is that we combine the conditional McKean-Vlasov equation with the associated stochastic Fokker-Planck partial integro-differential equation for the conditional law of the state. This leads to a Markovian system which can be handled by using a version of a Dynkin formula.
Our verification result is illustrated by finding the optimal time to sell in a market with common noise and jumps.