Recent zbMATH articles in MSC 35Chttps://zbmath.org/atom/cc/35C2021-06-15T18:09:00+00:00WerkzeugSurfaces of revolution associated with the kink-type solutions of the SIdV equation.https://zbmath.org/1460.353142021-06-15T18:09:00+00:00"Zhang, Guofei"https://zbmath.org/authors/?q=ai:zhang.guofei"He, Jingsong"https://zbmath.org/authors/?q=ai:he.jingsong"Wang, Lihong"https://zbmath.org/authors/?q=ai:wang.lihong"Mihalache, Dumitru"https://zbmath.org/authors/?q=ai:mihalache.dumitruSummary: In this paper, we study the evolution scenarios of surfaces of revolution associated with the kink-type solutions of an integrable equation, which is called the SIdV equation because of its scale-invariant property and relationship with the Korteweg-de Vries equation, where the kink-type solutions play the role of a metric. We put forward two kinds of evolution scenarios for surfaces of revolution associated with two types of kink-type metric (solution) and we study the key properties of these surfaces.Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations.https://zbmath.org/1460.353312021-06-15T18:09:00+00:00"Murphy, Jason"https://zbmath.org/authors/?q=ai:murphy.jason"Nakanishi, Kenji"https://zbmath.org/authors/?q=ai:nakanishi.kenji.1|nakanishi.kenji.2Summary: We consider nonlinear Schrödinger equations with either power-type or Hartree nonlinearity in the presence of an external potential. We show that for long-range nonlinearities, solutions cannot exhibit scattering to solitary waves or more general localized waves. This extends the well-known results concerning non-existence of non-trivial scattering states for long-range nonlinearities.Asymptotic behavior of solutions to the logarithmic diffusion equation with a linear source.https://zbmath.org/1460.350412021-06-15T18:09:00+00:00"Shimojo, Masahiko"https://zbmath.org/authors/?q=ai:shimojo.masahiko"Takáč, Peter"https://zbmath.org/authors/?q=ai:takac.peter"Yanagida, Eiji"https://zbmath.org/authors/?q=ai:yanagida.eijiSummary: We investigate the behavior of positive solutions to the Cauchy problem
\[
\begin{cases} \partial_t u =\partial_x^2(\log u)+u, & x\in\mathbb{R}, t>0, \\ \mathop{\lim}\limits_{x\rightarrow-\infty} \partial_x(\log u)=\alpha, \quad \mathop{\lim}\limits_{x\rightarrow+\infty}\partial_x(\log u)=\beta, \quad & t>0, \\ u(x,0)=u_0(x), & x\in\mathbb{R}, \end{cases}
\]
where \(\alpha,\beta \) are given positive constants and \(u_0(x)\) is a positive initial value. In the case of mass conservation, i.e., \(\int_{-\infty}^{\infty} u(x,t) dx \equiv \alpha + \beta \) for \(t\geq 0\), we show by an intersection number argument that the solution approaches a traveling wave as \(t\rightarrow \infty \). We then study the behavior in the case of mass expansion or extinction by using a transformation of variables. When the total mass is smaller, we show that extinction of the solution occurs in finite time and a rescaled solution converges to the traveling wave, whereas when the total mass is larger, the solution grows exponentially and a rescaled solution converges to a certain profile. Our results also include some log-concavity properties of solutions.Reduction of a damped, driven Klein-Gordon equation into a discrete nonlinear Schrödinger equation: justification and numerical comparison.https://zbmath.org/1460.353302021-06-15T18:09:00+00:00"Muda, Yuslenita"https://zbmath.org/authors/?q=ai:muda.yuslenita"Akbar, Fiki T."https://zbmath.org/authors/?q=ai:akbar.fiki-taufik"Kusdiantara, Rudy"https://zbmath.org/authors/?q=ai:kusdiantara.rudy"Gunara, Bobby E."https://zbmath.org/authors/?q=ai:gunara.bobby-eka"Susanto, Hadi"https://zbmath.org/authors/?q=ai:susanto.hadiSummary: We consider a discrete nonlinear Klein-Gordon equation with damping and external drive. Using a small amplitude ansatz, one usually approximates the equation using a damped, driven discrete nonlinear Schrödinger equation. Here, we show for the first time the justification of this approximation by finding the error bound using energy estimate. Additionally, we prove the local and global existence of the Schrödinger equation. Numerical simulations are performed that describe the analytical results. Comparisons between discrete breathers of the Klein-Gordon equation and discrete solitons of the discrete nonlinear Schrödinger equation are presented.The families of explicit solutions for the Hirota equation.https://zbmath.org/1460.353112021-06-15T18:09:00+00:00"Su, Ting"https://zbmath.org/authors/?q=ai:su.ting"Wang, Jia"https://zbmath.org/authors/?q=ai:wang.jiaThe authors present some explicit solutions for the Hirota equation derived from the Ablowitz-Kaup-Newell-Segur (AKNS) shallow water wave equation: A dark one soliton solution is obtained by using homogeneous balance for the Hirota equation; multiple soliton solutions and multiple singular solutions are derived by using the so-called Hirota bilinear form of the equation. Finally, the authors obtain one- and two-periodic solutions for the AKNS equations in the form of Riemann theta functions by employing the Hirota bilinear form of the AKNS equation. The asymptotic behaviour of the two periodic solutions is then indicated.
Reviewer: Catalin Popa (Iaşi)On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations.https://zbmath.org/1460.353212021-06-15T18:09:00+00:00"Comech, Andrew"https://zbmath.org/authors/?q=ai:comech.andrew"Cuccagna, Scipio"https://zbmath.org/authors/?q=ai:cuccagna.scipioSummary: We extend to a specific class of systems of nonlinear Schrödinger equations (NLS) the theory of asymptotic stability of ground states already proved for the scalar NLS. Here the key point is the choice of an adequate system of modulation coordinates and the novelty, compared to the scalar NLS, is the fact that the group of symmetries of the system is non-commutative.Nonlinear stability of periodic-wave solutions for systems of dispersive equations.https://zbmath.org/1460.350282021-06-15T18:09:00+00:00"Cristófani, Fabrício"https://zbmath.org/authors/?q=ai:cristofani.fabricio"Pastor, Ademir"https://zbmath.org/authors/?q=ai:pastor.ademirSummary: We prove the orbital stability of periodic traveling-wave solutions for systems of dispersive equations with coupled nonlinear terms. Our method is basically developed under two assumptions: one concerning the spectrum of the linearized operator around the traveling wave and another one concerning the existence of a conserved quantity with suitable properties. The method can be applied to several systems such as the Liu-Kubota-Ko system, the modified KdV system and a log-KdV type system.Periodic solutions of an age-structured epidemic model with periodic infection rate.https://zbmath.org/1460.353532021-06-15T18:09:00+00:00"Kang, Hao"https://zbmath.org/authors/?q=ai:kang.hao"Huang, Qimin"https://zbmath.org/authors/?q=ai:huang.qimin"Ruan, Shigui"https://zbmath.org/authors/?q=ai:ruan.shiguiSummary: In this paper we consider an age-structured epidemic model of the susceptible-exposed-infectious-recovered (SEIR) type. To characterize the seasonality of some infectious diseases such as measles, it is assumed that the infection rate is time periodic. After establishing the well-posedness of the initial-boundary value problem, we study existence of time periodic solutions of the model by using a fixed point theorem. Some numerical simulations are presented to illustrate the obtained results.Backward self-similar solutions for compressible Navier-Stokes equations.https://zbmath.org/1460.352562021-06-15T18:09:00+00:00"Germain, Pierre"https://zbmath.org/authors/?q=ai:germain.pierre"Iwabuchi, Tsukasa"https://zbmath.org/authors/?q=ai:iwabuchi.tsukasa"Léger, Tristan"https://zbmath.org/authors/?q=ai:leger.tristanHardy inequalities for the fractional powers of the Grushin operator.https://zbmath.org/1460.350072021-06-15T18:09:00+00:00"Song, Manli"https://zbmath.org/authors/?q=ai:song.manli"Tan, Jinggang"https://zbmath.org/authors/?q=ai:tan.jinggangSummary: We establish uncertainty principles and Hardy inequalities for the fractional Grushin operator, which are reduced to those inequalities for the fractional generalized sublaplacian. The key ingredients to obtain them are an explicit integral representation and a ground state representation for the fractional powers of generalized sublaplacian.On properties of solutions to the \(\alpha\)-harmonic equation.https://zbmath.org/1460.310022021-06-15T18:09:00+00:00"Li, Peijin"https://zbmath.org/authors/?q=ai:li.peijin"Rasila, Antti"https://zbmath.org/authors/?q=ai:rasila.antti"Wang, Zhi-Gang"https://zbmath.org/authors/?q=ai:wang.zhigangSummary: The aim of this paper is to establish properties of solutions to the \(\alpha \)-harmonic equations: \(\Delta_\alpha(f(z))=\partial z[(1-|z|^2)^{-\alpha}\bar{\partial}zf](z)=g(z)\), where \(g:\overline{\mathbb{D}}\to\mathbb{C}\) is a continuous function and \(\overline{\mathbb{D}}\) denotes the closure of the unit disc \(\mathbb{D}\) in the complex plane \(\mathbb{C}\). We obtain Schwarz type and Schwarz-Pick type inequalities for solutions to the \(\alpha\)-harmonic equation. In particular, for \(g\equiv 0\), solutions to the above equation are called \(\alpha\)-harmonic functions. We determine the necessary and sufficient conditions for an analytic function \(\psi\) to have the property that \(f\circ\psi\) is \(\alpha\)-harmonic function for any \(\alpha\)-harmonic function \(f\). Furthermore, we discuss the Bergman-type spaces on \(\alpha\)-harmonic functions.On nonlinear profile decompositions and scattering for an NLS-ODE model.https://zbmath.org/1460.353222021-06-15T18:09:00+00:00"Cuccagna, Scipio"https://zbmath.org/authors/?q=ai:cuccagna.scipio"Maeda, Masaya"https://zbmath.org/authors/?q=ai:maeda.masayaSummary: In this paper, we consider a Hamiltonian system combining a nonlinear Schrödinger equation (NLS) and an ordinary differential equation. This system is a simplified model of the NLS around soliton solutions. Following \textit{K. Nakanishi} [J. Math. Soc. Japan 69, No. 4, 1353--1401 (2017; Zbl 1383.35213)], we show scattering of \(L^2\) small \(H^1\) radial solutions. The proof is based on Nakanishi's framework and Fermi Golden Rule estimates on \(L^4\) in time norms.Global diffeomorphism of the Lagrangian flow-map for a Pollard-like internal water wave.https://zbmath.org/1460.353472021-06-15T18:09:00+00:00"Kluczek, Mateusz"https://zbmath.org/authors/?q=ai:kluczek.mateusz"Rodríguez-Sanjurjo, Adrián"https://zbmath.org/authors/?q=ai:rodriguez-sanjurjo.adrianSummary: In this article we provide an overview of a rigorous justification of the global validity of the fluid motion described by a new exact and explicit solution prescribed in terms of Lagrangian variables of the nonlinear geophysical equations. More precisely, the three-dimensional Lagrangian flow-map describing this exact and explicit solution is proven to be a global diffeomorphism from the labelling domain into the fluid domain. Then, the flow motion is shown to be dynamically possible.
For the entire collection see [Zbl 1432.35003].Representation of solutions of the Cauchy problem for a one-dimensional Schrödinger equation with a smooth bounded potential by quasi-Feynman formulae.https://zbmath.org/1460.810172021-06-15T18:09:00+00:00"Grishin, Denis V."https://zbmath.org/authors/?q=ai:grishin.denis-v"Pavlovskiy, Yan Yu."https://zbmath.org/authors/?q=ai:pavlovskiy.yan-yuMore about Q-ball with elliptical orbit.https://zbmath.org/1460.810502021-06-15T18:09:00+00:00"Hasegawa, Fuminori"https://zbmath.org/authors/?q=ai:hasegawa.fuminori"Hong, Jeong-Pyong"https://zbmath.org/authors/?q=ai:hong.jeong-pyong"Suzuki, Motoo"https://zbmath.org/authors/?q=ai:suzuki.motooSummary: Q-balls formed from the Affleck-Dine field have rich cosmological implications and have been extensively studied from both theoretical and simulational approaches. From the theoretical point of view, the exact solution of the Q-ball was obtained and it shows a circular orbit in the complex plane of the field value. In practice, however, it is reported that the Q-ball that appears after the Affleck-Dine mechanism has an \textit{elliptical} orbit, which carries larger energy per unit \(U(1)\) charge than the well-known solution with a circular orbit. We call them ``elliptical'' Q-balls. In this paper, we report the first detailed investigation of the properties of the elliptical Q-balls by 3D lattice simulation. The simulation results indicate that the elliptical Q-ball has an almost spherical spatial profile with no nodes, and we observed a highly elliptic orbit that cannot be described through small perturbations around the ground state Q-ball. Higher ellipticity leads to more excitation of the energy, whose relation is also derived as a dispersion relation. Finally, we derive two types of approximate solutions by extending the Gaussian approximation and considering the time-averaged equation of motion and we also show the consistency with the simulation results.Detection of small inhomogeneities via direct sampling method in transverse electric polarization.https://zbmath.org/1460.780122021-06-15T18:09:00+00:00"Park, Won-Kwang"https://zbmath.org/authors/?q=ai:park.won-kwangSummary: Various studies have confirmed the possibility of identifying the location of a set of small inhomogeneities via a direct sampling method; however, when their permeability differs from that of the background, their location cannot be satisfactorily identified. However, no theoretical explanation for this phenomenon has been verified. In this study, we demonstrate that the indicator function of the direct sampling method can be expressed by the Bessel function of order one of the first kind and explain why the exact locations of inhomogeneities cannot be identified. Numerical results with noisy data are exhibited to support our examination.Construction of a solitary wave solution of the nonlinear focusing Schrödinger equation outside a strictly convex obstacle in the \(L^2\)-supercritical case.https://zbmath.org/1460.353292021-06-15T18:09:00+00:00"Landoulsi, Oussama"https://zbmath.org/authors/?q=ai:landoulsi.oussamaSummary: We consider the focusing \(L^2\)-supercritical Schrödinger equation in the exterior of a smooth, compact, strictly convex obstacle \(\Theta\subset\mathbb{R}^3\). We construct a solution behaving asymptotically as a solitary wave on \(\mathbb{R}^3,\) for large times. When the velocity of the solitary wave is high, the existence of such a solution can be proved by a classical fixed point argument. To construct solutions with arbitrary nonzero velocity, we use a compactness argument similar to the one that was introduced by \textit{F. Merle} [Commun. Math. Phys. 129, No. 2, 223--240 (1990; Zbl 0707.35021)] to construct solutions of the NLS equation blowing up at several points together with a topological argument using Brouwer's theorem to control the unstable direction of the linearized operator at the soliton. These solutions are arbitrarily close to the scattering threshold given by a previous work of \textit{R. Killip} et al. [AMRX, Appl. Math. Res. Express 2016, No. 1, 146--180 (2016; Zbl 1345.35102)], which is the same as the one on the whole Euclidean space given by \textit{T. Duyckaerts} et al. [Math. Res. Lett. 15, No. 5--6, 1233--1250 (2008; Zbl 1171.35472)] and \textit{J. Holmer} and \textit{S. Roudenko} [Commun. Math. Phys. 282, No. 2, 435--467 (2008; Zbl 1155.35094)] in the radial case and by the previous authors with T. Duyckaerts in the non-radial case.On singular solutions of time-periodic and steady Stokes problems in a power cusp domain.https://zbmath.org/1460.352542021-06-15T18:09:00+00:00"Eismontaite, Alicija"https://zbmath.org/authors/?q=ai:eismontaite.alicija"Pileckas, Konstantin"https://zbmath.org/authors/?q=ai:pileckas.konstantinSummary: The time-periodic and steady Stokes problems with the boundary value having a nonzero flux are considered in the power cusp domains. The asymptotic expansion near the singularity point is constructed in order to reduce the problem to the case where the energy solution exists. The solution of the problem is found then as the sum of the asymptotic expansion and the term with finite dissipation of energy.Some comparisons between heterogeneous and homogeneous plates for nonlinear symmetric SH waves in terms of heterogeneous and nonlinear effects.https://zbmath.org/1460.350672021-06-15T18:09:00+00:00"Demirkuş, Dilek"https://zbmath.org/authors/?q=ai:demirkus.dilekSummary: In this article, the propagation of nonlinear shear horizontal waves for some comparisons between the heterogeneous and homogeneous plates is considered. It is assumed that one plate is made of up hyper-elastic, heterogeneous, isotropic, and generalized neo-Hookean materials, and the other consists of hyper-elastic, homogeneous, isotropic, and generalized neo-Hookean materials. Using the known solitary wave solutions, called bright and dark solitary wave solutions, to the nonlinear Schrödinger equation, these comparisons are made in terms of the heterogeneous and nonlinear effects. All numerical results, based on the asymptotic analyses in which the method of multiple scales is used, are graphically presented for the lowest dispersive symmetric branch of both dispersion relations.On the mathematical modelling of competitive invasive weed dynamics.https://zbmath.org/1460.921762021-06-15T18:09:00+00:00"Ramaj, Tedi"https://zbmath.org/authors/?q=ai:ramaj.tediSummary: We explore the dynamics of invasive weeds by partial differential equation (PDE) modelling and applying dynamical system and phase portrait techniques. We begin by applying the method of characteristics to a preexisting PDE model of the spreading of \textit{T. fluminensis}, an invasive weed which has been responsible for native forest depletion. We explore the system both at particular points in space and over all of space, in one dimension, as a function of time. Our model suggests that an increase in the rate of spread of the weed through space will increase the efficacy of control measures taken at the weed's spatial boundary. We then propose new competition models based on the previous model and explore the existence of travelling wave solutions. These models represent both the cases with (i) a competing native plant species which spreads through the forest and (ii) a non-mobile, established native plant species. In the former case, the model suggests that an increased mass-action coefficient between the competing species is sufficient and necessary for the transition of the forest into a state of coexistence. In the latter case, the result is not as strong: a sufficiently large rate of competition between the species excludes the possibility of native plant extinction and hence suggests that forest depletion will not occur, but does not imply coexistence. We perform some numerical simulations to support our analytic results. In all cases, we give a discussion on the physical and biological interpretations of our results. We conclude with some suggestions for future work and with a discussion of the advantages and disadvantages of the methods.Numerical solutions of the generalized equal width wave equation using the Petrov-Galerkin method.https://zbmath.org/1460.651412021-06-15T18:09:00+00:00"Bhowmik, Samir Kumar"https://zbmath.org/authors/?q=ai:bhowmik.samir-kumar"Karakoc, Seydi Battal Gazi"https://zbmath.org/authors/?q=ai:karakoc.seydi-battal-gaziIn this paper, a lumped Petrov-Galerkin method is constructed for the a generalized equal width wave (GEW) equation using the quadratic B-spline function as the element shape function and the linear B-spline function as the weight function. Error estimates for the semi-discrete and fully discrete solutions are derived. The performance of the method is demonstrated for three test problems involving a single solitary wave in which analytic solution is known and expanded it to investigate the interaction of two solitary waves and evolution of solitons where the analytic solutions are generally unknown during the interaction. The three invariants of the GEW, mass, momentum, and energy are preserved for a single soliton in the \(L_2\)- and \(L_\infty\) norms in long-term integration. A comparison with other schemes like cubic Galerkin, quintic collocation, and Petrov-Galerkin schemes demonstrates the effectiveness of the proposed method.
Reviewer: Bülent Karasözen (Ankara)Obtaining multisoliton solutions of the \((2+1)\)-dimensional Camassa-Holm system using Darboux transformations.https://zbmath.org/1460.370692021-06-15T18:09:00+00:00"Mao, Hui"https://zbmath.org/authors/?q=ai:mao.huiSummary: We construct and study Darboux transformations for the \((2+1)\)-dimensional Camassa-Holm system. We apply a reciprocal transformation that relates the \((2+1)\)-dimensional Camassa-Holm system and the linear system associated with the modified Kadomtsev-Petviashvili hierarchy. Using three Darboux transformation operators, we obtain three types of solutions for the \((2+1)\)-dimensional Camassa-Holm system, of which one is a multisoliton solution. In addition, we briefly discuss rational solutions.Explicit exact periodic wave solutions and their limit forms for a long waves-short waves model.https://zbmath.org/1460.350652021-06-15T18:09:00+00:00"He, Bin"https://zbmath.org/authors/?q=ai:he.bin"Meng, Qing"https://zbmath.org/authors/?q=ai:meng.qing.1It is considered the long waves -- short waves model
\[\begin{gathered}
A_t = 2\sigma(|B|^2)_x,\\
B_t = \imath B_{xx} - A_xB + \imath A^2B - 2\imath\sigma B|B|^2
\end{gathered}\]
with \(A(x,t)\) -- the amplitude of the long wave and \(B(x,t)\) -- the envelope of
the short wave. It is considered the traveling wave solution of the form
\[
A(x,t)=A(\xi), \quad B(x,t)=\phi(\xi)e^{\imath(\psi(\xi)-\omega t)}; \quad \xi = x-ct,
\]
where \(A(\xi)\), \(\phi(\xi)\), \(\psi(\xi)\) are real valued functions. By direct
computation it is obtained that
\[
A(\xi)=g - \frac{2\sigma}{c}\phi^2(\xi),
\]
\(g\) being the integration constant. \(\psi(\xi)\) and \(\phi(\xi)\) are solutions of a
system of ordinary differential equations which can be ``diagonalized'' as
\[
\begin{gathered}
\psi' = \frac{\sigma}{c}\phi^2 - \frac{1}{2}c, \\
\phi" = \rho(\phi^4 + \alpha\phi^2 + \beta)\phi.
\end{gathered}
\]
The paper discusses the qualitative properties of the planar dynamic system defined
by \((\phi,\phi')\) -- which has a first integral -- and obtains several analytic-computational results.
Reviewer: Vladimir Răsvan (Craiova)Existence and orbital stability of standing waves for the 1D Schrödinger-Kirchhoff equation.https://zbmath.org/1460.350302021-06-15T18:09:00+00:00"Natali, Fábio"https://zbmath.org/authors/?q=ai:natali.fabio-m-amorin"Cardoso, Eleomar"https://zbmath.org/authors/?q=ai:cardoso.eleomar-junSummary: In this paper we establish the orbital stability of standing wave solutions associated to the one-dimensional Schrödinger-Kirchhoff equation. The presence of a mixed term gives us more dispersion, and consequently, a different scenario for the stability of solitary waves in contrast with the corresponding nonlinear Schrödinger equation. For periodic waves, we exhibit two explicit solutions and prove the orbital stability in the energy space.Spreading speed of the periodic Lotka-Volterra competition model.https://zbmath.org/1460.352012021-06-15T18:09:00+00:00"Liu, Xiaolin"https://zbmath.org/authors/?q=ai:liu.xiaolin"Ouyang, Zigen"https://zbmath.org/authors/?q=ai:ouyang.zigen"Huang, Zhe"https://zbmath.org/authors/?q=ai:huang.zhe"Ou, Chunhua"https://zbmath.org/authors/?q=ai:ou.chunhuaThis paper is concerned with the minimal speed (spreading speed) selection mechanism of traveling waves to a periodic diffusive Lotka-Volterra model with monostable nonlinearity. For the Lotka-Volterra competition model with constant coefficients, the minimal speed determinacy of traveling waves has been investigated by \textit{A. Alhasanat} and \textit{C. Ou} [J. Differ. Equations 266, No. 11, 7357--7378 (2019; Zbl 1408.35067)]. The main method is the upper-lower solution technique.
Taking into account the influence of seasonal changes, the authors studied a diffusive Lotka-Volterra model with the periodic coefficients. The method of upper and lower solutions pair is also applied to establish the existence of traveling waves and the minimal speed determinacy. Comparing with the model with constant coefficients, it is not easy to find upper (or lower) solutions for both equations simultaneously. By constructing new pairs of upper and lower solutions that are totally different from the classical ones, the authors established new results on both the linear and nonlinear speed selection mechanism. In addition, they showed that the significant nature of the nonlinear selection is to find a lower solution pair with the density of the first species decaying in a faster rate. This provides a practical way to find a bound estimate for the spreading speed as well as when the nonlinear selection is realized.
Reviewer: Guobao Zhang (Lanzhou)On competition models under Allee effect: asymptotic behavior and traveling waves.https://zbmath.org/1460.350292021-06-15T18:09:00+00:00"Feng, Wei"https://zbmath.org/authors/?q=ai:feng.wei"Freeze, Michael"https://zbmath.org/authors/?q=ai:freeze.michael"Lu, Xin"https://zbmath.org/authors/?q=ai:lu.xinSummary: In this article, we study a reaction-diffusion model on infinite spatial domain for two competing biological species (\(u\) and \(v\)). Under one-side Allee effect on \(u\)-species, the model demonstrates complexity on its coexistence and \(u\)-dominance steady states. The conditions for persistence, permanence and competitive exclusion of the species are obtained through analysis on asymptotic behavior of the solutions and stability of the steady states, including the attraction regions and convergent rates depending on the biological parameters. When the Allee effect constant \(K\) is large relative to other biological parameters, the asymptotic stability of the \(v\)-dominance state \((0,1)\) indicates the competitive exclusion of the \(u\)-species. Applying upper-lower solution method, we further prove that for a family of wave speeds with specific minimum wave speed determined by several biological parameters (including the magnitude of the \(u\)-dominance states), there exist traveling wave solutions flowing from the \(u\)-dominance states to the \(v\)-dominance state. The asymptotic rates of the traveling waves at \(\xi\rightarrow\mp\infty\) are also explicitly calculated. Finally, numerical simulations are presented to illustrate the theoretical results and population dynamics of coexistence or dominance-shifting.Existence of traveling waves for a nonlocal dispersal SIR epidemic model with treatment.https://zbmath.org/1460.921922021-06-15T18:09:00+00:00"Deng, Dong"https://zbmath.org/authors/?q=ai:deng.dong"Li, Jianzhong"https://zbmath.org/authors/?q=ai:li.jianzhong"Zhang, Dongpei"https://zbmath.org/authors/?q=ai:zhang.dongpeiSummary: This paper is concerned with the existence and nonexistence of traveling wave solutions for a nonlocal dispersal epidemic model with treatment. The existence of traveling wave solutions is established by Schauder's fixed point theorem, while the nonexistence of traveling wave solutions is proved by two-sided Laplace transform. From the results, we conclude the minimal wave speed, which is an important threshold to predict how fast the disease invades. Compared with the work in [\textit{T. Zhang} and \textit{W. Wang}, J. Math. Anal. Appl. 419, No. 1, 469--495 (2014; Zbl 1295.35175)], we obtain more accurate results about the existence and nonexistence of nontrivial traveling wave solutions. We prove that when the basic reproduction number \(\mathcal{R}_0 > 1\), there exists a critical number \(c_1^\ast > 0\) such that for each \(c > c_1^\ast \), the system has a nontrivial traveling wave solution with speed \(c\), while for \(0 < c < c_1^\ast\) the system admits no nontrivial traveling wave solution. When \(\mathcal{R}_0 < 1\), we show that there exists no nontrivial traveling wave solution. In addition, based on [\textit{C.-C. Wu}, J. Differ. Equations 262, No. 1, 272--282 (2017; Zbl 1387.34094)], we obtain the existence of traveling waves with the critical speed \(c = c_1^\ast\) under certain conditions.Chaotic study on a multibody interacting particle system with trajectory of variable curvature radius.https://zbmath.org/1460.820142021-06-15T18:09:00+00:00"Wang, Yu-Qing"https://zbmath.org/authors/?q=ai:wang.yuqing"Lin, Sen"https://zbmath.org/authors/?q=ai:lin.sen"Yang, Xiao-Dong"https://zbmath.org/authors/?q=ai:yang.xiaodong"Hong, Fang"https://zbmath.org/authors/?q=ai:hong.fang"Wang, Bing-Hong"https://zbmath.org/authors/?q=ai:wang.binghongSummary: Multibody interacting particle system is one of the most important driven-diffusive systems, which can well describe stochastic dynamics of self-driven particles unidirectionally updating along one-dimensional discrete lattices controlled by hard-core exclusions. Such determined nonlinear dynamic system shows the chaos, a complicated motion that is similar to random and can't determine future states according to given initial conditions. Derived from the study of randomness in the framework of determinism, chaos effectively unifies determinacy and randomness of nonlinear physics, which also embodies the randomness and uncertainty of multi-body particle motions. Besides, chaos often presents complex ordered phenomena like aperiodic ordered motion state etc., which indicates it's a stable coordination of local random and global mode. Different with previous work, a mesoscopic multibody interacting particle system considering the trajectory of variable curvature radius is proposed, which is more reliable to depict true driven-diffusive systems. Nonlinear dynamical master equations are established. Linear and nonlinear stability analyses are performed to test system robustness, which lead to linear stable region, metastable region, unstable region, triangular wave, solitary wave and kink-antikink wave. Complete numerical simulations under quantitatively changing characteristic order parameters are performed to reveal intrinsic chaotic dynamics mechanisms. By comparing fruitful chaotic patterns, double periodic convergence in stable limit cycles is observed due to attractors folding. System states tend to be limit cycles with the passage of time. The longer time is, the higher density is near attractors. The number of main components of attractors and the shape of them are found to drastically change with these parameters. Hereafter, the chaotic formation mechanism is explained by performing Fourier spectrums analyses. High and low frequency elements are obtained, whose module lengths are found to change periodically with time and nonperiodically with space. Our results show the sensitivity of the chaotic system to initial conditions, which mean slight changes of initial states in one part of system can lead to disproportionate consequences in other parts. The sensitivity is directly related to uncertainty and unpredictability. Moreover, calculations also show that three kinds of attractors (namely, point attractor, limit-cycle attractor and strange attractor) exist in the proposed chaotic system, which control particle motions. The first two attractors are convergence attractors playing a limiting role, which lead to static and balanced features of system. Oppositely, strange attractor makes the system deviate from regions of convergent attractors and creates unpredictability by inducing the vitality of the system and turning it into a non-preset mode. Furthermore, interactions between convergence attractors and strange attractors trigger such locally fruitful modes. This work will be helpful to understand mesoscopic dynamics, stochastic dynamics and non-equilibrium dynamical behaviors of multibody interacting particle systems.Parabolic equations involving Laguerre operators and weighted mixed-norm estimates.https://zbmath.org/1460.350662021-06-15T18:09:00+00:00"Fan, Huiying"https://zbmath.org/authors/?q=ai:fan.huiying"Ma, Tao"https://zbmath.org/authors/?q=ai:ma.taoSummary: In this paper, we study evolution equation \(\partial_tu=-L_\alpha u+f\) and the corresponding Cauchy problem, where \(L_\alpha\) represents the Laguerre operator \(L_\alpha=\frac{1}{2}(-\frac{d^2}{dx^2}+x^2+\frac{1}{x^2}(\alpha^2-\frac{1}{4}))\), for every \(\alpha\geq-\frac{1}{2}\). We get explicit pointwise formulas for the classical solution and its derivatives by virtue of the parabolic heat-diffusion semigroup \(\{e^{-\tau(\partial_t+L_\alpha)}\}_{\tau>0}\). In addition, we define the Poisson operator related to the fractional power \((\partial_t+L_\alpha)^s\) and reveal weighted mixed-norm estimates for revelent maximal operators.Representations of the solutions of the first-order elliptic and hyperbolic systems via harmonic and wave functions Respectively.https://zbmath.org/1460.350642021-06-15T18:09:00+00:00"Tokibetov, J."https://zbmath.org/authors/?q=ai:tokibetov.j-a"Abduakhitova, G."https://zbmath.org/authors/?q=ai:abduakhitova.g-e"Assadi, A."https://zbmath.org/authors/?q=ai:assadi.amir-hSummary: In this paper representations of the solutions of all first-order elliptic and hyperbolic systems in three-dimensional and four-dimensional spaces are obtained through the derivatives of harmonic and wave functions, respectively.Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity. II: Supercritical blow-up profiles.https://zbmath.org/1460.353282021-06-15T18:09:00+00:00"Holmer, Justin"https://zbmath.org/authors/?q=ai:holmer.justin"Liu, Chang"https://zbmath.org/authors/?q=ai:liu.chang.1Summary: We consider the 1D nonlinear Schrödinger equation (NLS) with focusing point nonlinearity,
\[
i\partial_t\psi+\partial_x^2\psi+\delta|\psi|^{p-1}\psi=0,\tag{(0.1)}
\]
where \(\delta=\delta(x)\) is the delta function supported at the origin. In the \(L^2\) supercritical setting \(p>3\), we construct self-similar blow-up solutions belonging to the energy space \(L_x^\infty\cap\dot H_x^1\). This is reduced to finding outgoing solutions of a certain stationary profile equation. All outgoing solutions to the profile equation are obtained by using parabolic cylinder functions (Weber functions) and solving the jump condition at \(x=0\) imposed by the \(\delta\) term in (0.1). This jump condition is an algebraic condition involving gamma functions, and existence and uniqueness of solutions is obtained using the intermediate value theorem and formulae for the digamma function. We also compute the form of these outgoing solutions in the slightly supercritical case \(0<p-3\ll 1\) using the log Binet formula for the gamma function and steepest descent method in the integral formulae for the parabolic cylinder functions.
For part I, see [the authors, J. Math. Anal. Appl. 483, No. 1, Article ID 123522, 20 p. (2020; Zbl 1436.35290)].Time decay of the solution to the Cauchy problem for a three-dimensional model of nonsimple thermoelasticity.https://zbmath.org/1460.353422021-06-15T18:09:00+00:00"Łazuka, Jarosław"https://zbmath.org/authors/?q=ai:lazuka.jaroslawThe paper is devoted to thermoelasticity of non-simple materials in a three-dimensional space. The mathematical model is described by a system of partial differential equations of fourth order. After defining a suitable evolution equation, the existence of the solution to the Cauchy problem is proven by applying semigroup methods. An asymptotic analysis of the solution is developed. The explicit Fourier representation of the solution is derived. By employing Sobolev, Bessel and Besov spaces and by applying the interpolation method, the author shows the \(L^p-L^q\) time decay estimates for the solution.
Reviewer: Adina Chirila (Braşov)Some recent developments in the theory and applications of reaction-diffusion waves.https://zbmath.org/1460.350032021-06-15T18:09:00+00:00"Volpert, Vitaly"https://zbmath.org/authors/?q=ai:volpert.vitaly-aSummary: Some recent developments in the theory and applications of travelling wave solutions of parabolic equations are discussed. These results continue the works by Aizik Volpert on index and solvability conditions of elliptic problems, topological degree, spectral properties and bifurcations, wave existence and stability.Non-linear anti-symmetric shear motion: a comparative study of non-homogeneous and homogeneous plates.https://zbmath.org/1460.353412021-06-15T18:09:00+00:00"Demirkuş, Dilek"https://zbmath.org/authors/?q=ai:demirkus.dilekSummary: In this article, the non-linear anti-symmetric shear motion for some comparative studies between the non-homogeneous and homogeneous plates, having two free surfaces with stress-free, is considered. Assuming that one plate contains hyper-elastic, non-homogeneous, isotropic, and generalized neo-Hookean materials and the other one consists of hyper-elastic, homogeneous, isotropic, and generalized neo-Hookean materials. Using the method of multiple scales, the self-modulation of the non-linear anti-symmetric shear motion in these plates, as the non-linear Schrödinger (NLS) equations, can be given. Using the known solitary wave solutions, called bright and dark solitary wave solutions, to NLS equations, these comparative studies in terms of the non-homogeneous and non-linear effects are made. All numerical results, based on the asymptotic analyses, are graphically presented for the lowest anti-symmetric branches of both dispersion relations, including the deformation fields of plates.Invading and receding sharp-fronted travelling waves.https://zbmath.org/1460.922322021-06-15T18:09:00+00:00"El-Hachem, Maud"https://zbmath.org/authors/?q=ai:el-hachem.maud"McCue, Scott W."https://zbmath.org/authors/?q=ai:mccue.scott-william"Simpson, Matthew J."https://zbmath.org/authors/?q=ai:simpson.matthew-jSummary: Biological invasion, whereby populations of motile and proliferative individuals lead to moving fronts that invade vacant regions, is routinely studied using partial differential equation models based upon the classical Fisher-KPP equation. While the Fisher-KPP model and extensions have been successfully used to model a range of invasive phenomena, including ecological and cellular invasion, an often-overlooked limitation of the Fisher-KPP model is that it cannot be used to model biological recession where the spatial extent of the population decreases with time. In this work, we study the \textit{Fisher-Stefan} model, which is a generalisation of the Fisher-KPP model obtained by reformulating the Fisher-KPP model as a moving boundary problem. The nondimensional Fisher-Stefan model involves just one parameter, \( \kappa \), which relates the shape of the density front at the moving boundary to the speed of the associated travelling wave, \(c\). Using numerical simulation, phase plane and perturbation analysis, we construct approximate solutions of the Fisher-Stefan model for both slowly invading and receding travelling waves, as well as for rapidly receding travelling waves. These approximations allow us to determine the relationship between \(c\) and \(\kappa\) so that commonly reported experimental estimates of \(c\) can be used to provide estimates of the unknown parameter \(\kappa \). Interestingly, when we reinterpret the Fisher-KPP model as a moving boundary problem, many overlooked features of the classical Fisher-KPP phase plane take on a new interpretation since travelling waves solutions with \(c < 2\) are normally disregarded. This means that our analysis of the Fisher-Stefan model has both practical value and an inherent mathematical value.A posteriori error estimates for self-similar solutions to the Euler equations.https://zbmath.org/1460.352682021-06-15T18:09:00+00:00"Bressan, Alberto"https://zbmath.org/authors/?q=ai:bressan.alberto"Shen, Wen"https://zbmath.org/authors/?q=ai:shen.wenSummary: The main goal of this paper is to analyze a family of ``simplest possible'' initial data for which, as shown by numerical simulations, the incompressible Euler equations have multiple solutions. We take here a first step toward a rigorous validation of these numerical results. Namely, we consider the system of equations corresponding to a self-similar solution, restricted to a bounded domain with smooth boundary. Given an approximate solution obtained via a finite dimensional Galerkin method, we establish a posteriori error bounds on the distance between the numerical approximation and the exact solution having the same boundary data.Asymptotic expansion of low-energy excitations for weakly interacting bosons.https://zbmath.org/1460.811232021-06-15T18:09:00+00:00"Boßmann, Lea"https://zbmath.org/authors/?q=ai:bossmann.lea"Petrat, Sören"https://zbmath.org/authors/?q=ai:petrat.soren"Seiringer, Robert"https://zbmath.org/authors/?q=ai:seiringer.robertSummary: We consider a system of \(N\) bosons in the mean-field scaling regime for a class of interactions including the repulsive Coulomb potential. We derive an asymptotic expansion of the low-energy eigenstates and the corresponding energies, which provides corrections to Bogoliubov theory to any order in \(1/N\).Nonlinear waves in a quintic FitzHugh-Nagumo model with cross diffusion: fronts, pulses, and wave trains.https://zbmath.org/1460.352022021-06-15T18:09:00+00:00"Zemskov, Evgeny P."https://zbmath.org/authors/?q=ai:zemskov.evgeny-p"Tsyganov, Mikhail A."https://zbmath.org/authors/?q=ai:tsyganov.mikhail-a"Kassner, Klaus"https://zbmath.org/authors/?q=ai:kassner.klaus"Horsthemke, Werner"https://zbmath.org/authors/?q=ai:horsthemke.wernerSummary: We study a tristable piecewise-linear reaction-diffusion system, which approximates a quintic FitzHugh-Nagumo model, with linear cross-diffusion terms of opposite signs. Basic nonlinear waves with oscillatory tails, namely, fronts, pulses, and wave trains, are described. The analytical construction of these waves is based on the results for the bistable case [the first author et al., ``Wavy fronts and speed bifurcation in excitable systems with cross diffusion'', Phys. Rev. E (3) 77, No. 3, Article ID 036219, 6 p. (2008; \url{doi:10.1103/PhysRevE.77.036219}); ``Oscillatory pulses and wave trains in a bistable reaction-diffusion system with cross diffusion'', ibid. 95, No. 1, Article ID 012203, 9 p. (2017; \url{10.1103/PhysRevE.95.012203})] for fronts and for pulses and wave trains, respectively]. In addition, these constructions allow us to describe novel waves that are specific to the tristable system. Most interesting is the pulse solution with a zigzag-shaped profile, the bright-dark pulse, in analogy with optical solitons of similar shapes. Numerical simulations indicate that this wave can be stable in the system with asymmetric thresholds; there are no stable bright-dark pulses when the thresholds are symmetric. In the latter case, the pulse splits up into a tristable front and a bistable one that propagate with different speeds. This phenomenon is related to a specific feature of the wave behavior in the tristable system, the multiwave regime of propagation, i.e., the coexistence of several waves with different profile shapes and propagation speeds at the same values of the model parameters.
{\copyright 2021 American Institute of Physics}Modulation instability and optical solitons of Radhakrishnan-Kundu-Lakshmanan model.https://zbmath.org/1460.780232021-06-15T18:09:00+00:00"Raza, Nauman"https://zbmath.org/authors/?q=ai:raza.nauman"Javid, Ahmad"https://zbmath.org/authors/?q=ai:javid.ahmadSummary: This paper studies the solitons of Radhakrishnan-Kundu-Lakshmanan (RKL) model with power law nonlinearity. The modified simple equation method and \(\exp(-\varphi(q))\) method are presented as integration mechanisms. Dark, bright, singular and periodic soliton solutions are extracted as well as the constraint conditions for their existence. A prized discussion on the stability of these soliton profiles on the basis of index of the power law nonlinearity is also carried out with the help of physical description of solutions. The integration techniques have been proved to be extremely efficient and robust to find new optical solitary wave solutions for various nonlinear evolution equations describing optical pulse propagation. Moreover, using linear stability analysis, modulation instability of the RKL model is studied. Different effects contributing to the modulation instability spectrum gain are analyzed.Dynamical behavior of traveling wave solutions for a \((2+1)\)-dimensional Bogoyavlenskii coupled system.https://zbmath.org/1460.370682021-06-15T18:09:00+00:00"Leta, Temesgen Desta"https://zbmath.org/authors/?q=ai:leta.temesgen-desta"Liu, Wenjun"https://zbmath.org/authors/?q=ai:liu.wenjun"Achab, Abdelfattah El"https://zbmath.org/authors/?q=ai:el-achab.abdelfattah"Rezazadeh, Hadi"https://zbmath.org/authors/?q=ai:rezazadeh.hadi"Bekir, Ahmet"https://zbmath.org/authors/?q=ai:bekir.ahmetSummary: In this paper, we applied some computational tools, namely the modified extended tanh method via a Riccati equation, the general \(\mathrm{Exp}_a\)-function method and the bifurcation methods to study a nonlinear \((2+1)\)-dimensional Bogoyavlenskii coupled system in thin-film ferroelectric medium to construct exact traveling wave solutions. By applying a classical wave transformation we obtained an ordinary differential equations. As a result, some new traveling wave solutions are obtained including hyperbolic, trigonometric, exponential functions and rational forms. If the parameters take specific values, then the periodic wave, solitary waves, kink and anti-kink wave solutions are derived from the traveling waves. Also, we draw 2D and 3D graphics of exact solutions for the special cases of these nonlinear equations by the help of programming language Maple.Quantum Hamiltonians with weak random abstract perturbation. II: Localization in the expanded spectrum.https://zbmath.org/1460.352402021-06-15T18:09:00+00:00"Borisov, Denis"https://zbmath.org/authors/?q=ai:borisov.denis-i"Täufer, Matthias"https://zbmath.org/authors/?q=ai:taufer.matthias"Veselić, Ivan"https://zbmath.org/authors/?q=ai:veselic.ivanSummary: We consider multi-dimensional Schrödinger operators with a weak random perturbation distributed in the cells of some periodic lattice. In every cell the perturbation is described by the translate of a fixed abstract operator depending on a random variable. The random variables, indexed by the lattice, are assumed to be independent and identically distributed according to an absolutely continuous probability density. A small global coupling constant tunes the strength of the perturbation. We treat analogous random Hamiltonians defined on multi-dimensional layers, as well. For such models we determine the location of the almost sure spectrum and its dependence on the global coupling constant. In this paper we concentrate on the case that the spectrum expands when the perturbation is switched on. Furthermore, we derive a Wegner estimate and an initial length scale estimate, which together with Combes-Thomas estimate allow to invoke the multi-scale analysis proof of localization. We specify an energy region, including the bottom of the almost sure spectrum, which exhibits spectral and dynamical localization. Due to our treatment of general, abstract perturbations our results apply at once to many interesting examples both known and new.
For Part I, see [the first author et al., Ann. Henri Poincaré 17, No. 9, 2341--2377 (2016; Zbl 1348.82039)].