Recent zbMATH articles in MSC 70Hhttps://zbmath.org/atom/cc/70H2024-05-13T19:39:47.825584ZWerkzeugStationary coupled KdV hierarchies and related Poisson structureshttps://zbmath.org/1532.354072024-05-13T19:39:47.825584Z"Fordy, Allan P."https://zbmath.org/authors/?q=ai:fordy.allan-p"Huang, Qing"https://zbmath.org/authors/?q=ai:huang.qingSummary: In this paper we continue our analysis of the stationary flows of \(M\) component, coupled KdV (cKdV) hierarchies and their modifications. We describe the general structure of the \(t_1\) and \(t_2\) flows, using the case \(M = 3\) as our main example. One of our stationary reductions gives \(N\) degrees of freedom, superintegrable systems. When \(N = 1\) (for \(t_1)\) and \(N = 2\) (for \(t_2)\), we have Poisson maps, which give multi-Hamiltonian representations of the flows. We discuss the general structure of these Poisson tensors and give explicit forms for the case \(M = 3\). In this case there are 3 modified hierarchies, each with 4 Poisson brackets.
The stationary \(t_2\) flow (for \(N = 2)\) is separable in parabolic coordinates. Each Poisson bracket has rank 4, with \(M + 1\) Casimirs. The \(4 \times 4\) ``core'' of the Poisson tensors are nonsingular and related by a ``recursion operator''. The remaining part of each tensor is built out of the two commuting Hamiltonian vector fields, depending upon the specific Casimirs. The Poisson brackets are generalised to include the entire class of potential, separable in parabolic coordinates. The Jacobi identity imposes specific dependence on some parameters, representing the Casimirs of the extended canonical bracket. This general case is no longer a stationary cKdV flow, with Lax representation. We give a recursive procedure for constructing the Lax representation of the stationary flow for all values of \(M\), \textit{without} having to go through the stationary reduction.Generalized Darboux transformation for nonlinear Schrödinger system on general Hermitian symmetric spaces and rogue wave solutionshttps://zbmath.org/1532.354142024-05-13T19:39:47.825584Z"Asadi, Esmaeel"https://zbmath.org/authors/?q=ai:asadi.esmaeel"Riaz, H. W. A."https://zbmath.org/authors/?q=ai:riaz.h-wajahat-ahmed.2"Ganjkhanloo, Mohammad Ali"https://zbmath.org/authors/?q=ai:ganjkhanloo.mohammad-aliSummary: In this paper, a generalized Darboux transformation is obtained for Fordy-Kulish NLS (nonlinear Schrödinger) systems on general Hermitian symmetric spaces in order to rigorously obtain rogue wave solutions for these systems. In particular, we express the generalized algebraic relations in a simple and elegant compact form. As an illustration, we derive multi-soliton, breather-type and mainly rogue wave solutions of triangular patterns for single- and multi-component NLS systems on \(CP^1\) and \(SP(2)/U(2),\) respectively. We also analyze the modulation instability of proper plane wave solutions. In order to get visual intuition for the dynamics of the result and solutions for the running examples, the associated simulations of profiles are furnished as well.The time-fractional generalized Z-K equation: analysis of Lie group, similarity reduction, conservation laws, and explicit solutionshttps://zbmath.org/1532.354742024-05-13T19:39:47.825584Z"AL-Denari, Rasha B."https://zbmath.org/authors/?q=ai:al-denari.rasha-b"Ahmed, Engy. A."https://zbmath.org/authors/?q=ai:ahmed.engy-a"Tharwat, Mohammed M."https://zbmath.org/authors/?q=ai:tharwat.mohammed-m(no abstract)Symplectic and inverse spectral geometry of integrable systems: a glimpse and open problemshttps://zbmath.org/1532.370022024-05-13T19:39:47.825584Z"Pelayo, Álvaro"https://zbmath.org/authors/?q=ai:pelayo.alvaroSummary: We first give a glimpse of finite dimensional classical integrable Hamiltonian systems from the point of view of symplectic geometry and briefly discuss their quantum counterparts, with an emphasis on recent progress on inverse spectral geometry. Then we propose several open problems about the geometry, topology and dynamics of these systems. The problems are largely motivated by the works of a number of authors, including Arnold, Atiyah, Colin de Verdière, Delzant, Duistermaat, Eliasson, Fomenko, Guillemin, Kolmogorov, Kostant, Moser and Sternberg.On Dirac structures admitting a variational approachhttps://zbmath.org/1532.370512024-05-13T19:39:47.825584Z"Cosserat, Oscar"https://zbmath.org/authors/?q=ai:cosserat.oscar"Kotov, Alexei"https://zbmath.org/authors/?q=ai:kotov.alexei"Laurent-Gengoux, Camille"https://zbmath.org/authors/?q=ai:laurent-gengoux.camille"Ryvkin, Leonid"https://zbmath.org/authors/?q=ai:ryvkin.leonid"Salnikov, Vladimir"https://zbmath.org/authors/?q=ai:salnikov.vladimirSummary: We discuss the notion of horizontal cohomology for Dirac structures and, more generally, Lie algebroids. We then use this notion {to describe the condition allowing} a variational formulation of Dirac dynamics.Linearity of minimally superintegrable systems in a static electromagnetic fieldhttps://zbmath.org/1532.370532024-05-13T19:39:47.825584Z"Bertrand, S."https://zbmath.org/authors/?q=ai:bertrand.sylvain|bertrand.sebastien"Nucci, M. C."https://zbmath.org/authors/?q=ai:nucci.maria-claraThe manuscript deals with superintegrable systems in a static electromagnetic field. The focus is exploring hidden symmetries leading to linearizable equations. Fifteen nonlinear minimally superintegrable systems are analyzed and linearized by means of their hidden symmetries.
The paper contains an extensive analysis, including a plethora of cases and subcases studied in detail. The results align with the conjecture that all three-dimensional minimally superintegrable systems are linearizable, employing hidden symmetries.
Reviewer: J. M. Hoff da Silva (Guaratinguetá)Koenigs theorem and superintegrable Liouville metricshttps://zbmath.org/1532.370542024-05-13T19:39:47.825584Z"Valent, Galliano"https://zbmath.org/authors/?q=ai:valent.gallianoThe author gives a complete classification of all two-dimensional strictly Riemannian metrics of Liouville type whose geodesic Hamiltonian admits three functionally independent first integrals all quadratic in the momenta, i.e., with geodesic flow quadratically superintegrable. The cases when one or two of the first integrals can be reduced to linear in the momenta ones are excluded. Liouville metrics are the two-dimensional metrics admitting complete integrals of the geodesic Hamilton-Jacobi equation additively separable in some coordinate system, hence admitting at least one first integral which is quadratic in the momentum and independent from the Hamiltonian: the Liouville first integral. First, the subcase of Koenigs metrics, the ones corresponding to surfaces of revolution, is considered and the Koenigs classification of quadratically superintegrable metrics is obtained by a direct proof. Then, the more general case of Liouville metrics is considered. All the possible solutions are obtained by searching for the most general quadratic first integral commuting with both the Hamiltonian and the Liouville quadratic first integral. The integration is performed by taking into account the fact that two of the components of the third first integral must be harmonically conjugate. Eight normal forms of the metric are determined, and the three-dimensional algebra of first integrals of each form is explicitly described. Coupling-constant metamorphosis, a transformation between Hamiltonian systems preserving integrability and superintegrability, is employed to recover some cases of quadratically superintegrable systems known in the literature.
Reviewer: Giovanni Rastelli (Vercelli)Practical perspectives on symplectic accelerated optimizationhttps://zbmath.org/1532.370682024-05-13T19:39:47.825584Z"Duruisseaux, Valentin"https://zbmath.org/authors/?q=ai:duruisseaux.valentin"Leok, Melvin"https://zbmath.org/authors/?q=ai:leok.melvinSummary: Geometric numerical integration has recently been exploited to design symplectic accelerated optimization algorithms by simulating the Bregman Lagrangian and Hamiltonian systems from the variational framework introduced by \textit{A. Wibisono} et al. [Proc. Natl. Acad. Sci. USA 113, No. 47, E7351--E7358 (2016; Zbl 1404.90098)]. In this paper, we discuss practical considerations which can significantly boost the computational performance of these optimization algorithms and considerably simplify the tuning process. In particular, we investigate how momentum restarting schemes ameliorate computational efficiency and robustness by reducing the undesirable effect of oscillations and ease the tuning process by making time-adaptivity superfluous. We also discuss how temporal looping helps avoiding instability issues caused by numerical precision, without harming the computational efficiency of the algorithms. Finally, we compare the efficiency and robustness of different geometric integration techniques and study the effects of the different parameters in the algorithms to inform and simplify tuning in practice. From this paper emerge symplectic accelerated optimization algorithms whose computational efficiency, stability and robustness have been improved, and which are now much simpler to use and tune for practical applications.On the weighted inertia-energy approach to forced wave equationshttps://zbmath.org/1532.490202024-05-13T19:39:47.825584Z"Mainini, Edoardo"https://zbmath.org/authors/?q=ai:mainini.edoardo"Percivale, Danilo"https://zbmath.org/authors/?q=ai:percivale.daniloIn this paper, the authors study the weighted inertia-energy approach to forced wave equations. More precisely, they show the convergence of minimizers of weighted inertia-energy functionals to solutions of initial value problems for a class of nonlinear wave equations.
Reviewer: Savin Treanţă (Bucureşti)The orbital stability analysis of pendulum oscillations of a heavy rigid body with a fixed point under the Goryachev-Chaplygin conditionhttps://zbmath.org/1532.700102024-05-13T19:39:47.825584Z"Bardin, B. S."https://zbmath.org/authors/?q=ai:bardin.boris-sabirovich"Maksimov, B. A."https://zbmath.org/authors/?q=ai:maksimov.b-aSummary: We consider the motion of a heavy rigid body with a fixed point in a uniform gravitational field under the assumption that the principal moments of inertia satisfy the Goryachev-Chaplygin condition at the fixed point. We study the orbital stability problem for small pendulum oscillations of the body. We derive the equations of perturbed motion and reduce the problem to the study of the stability of the equilibrium position of a second order \(2 \pi \)-periodic Hamiltonian system. We find regions of parametric resonance and perform the nonlinear analysis of orbital stability outside these regions.Networks of periodic orbits in the circular restricted three-body problem with first order post-Newtonian termshttps://zbmath.org/1532.700162024-05-13T19:39:47.825584Z"Zotos, Euaggelos E."https://zbmath.org/authors/?q=ai:zotos.euaggelos-e"Papadakis, K. E."https://zbmath.org/authors/?q=ai:papadakis.konstantinos-e"Suraj, Md Sanam"https://zbmath.org/authors/?q=ai:suraj.md-sanam"Mittal, Amit"https://zbmath.org/authors/?q=ai:mittal.amit"Aggarwal, Rajiv"https://zbmath.org/authors/?q=ai:aggarwal.rajivSummary: The motivation of this article is to numerically investigate the orbital dynamics of the planar post-Newtonian circular restricted problem of three bodies. By numerically integrating several large sets of initial conditions of orbits we obtain the basins of escape. Additionally, we determine the influence of the transition parameter on the orbital structure of the system, as well as on the families of simple symmetric periodic orbits. The networks and the stability of the symmetric periodic orbits are revealed, while the corresponding critical periodic solutions are also identified. The parametric evolution of the horizontal and the vertical stability of the periodic orbits are also monitored, as a function of the transition parameter.Computer-assisted proofs of existence of KAM tori in planetary dynamical models of \(\upsilon\)-And \(\mathbf{b}\)https://zbmath.org/1532.700222024-05-13T19:39:47.825584Z"Mastroianni, Rita"https://zbmath.org/authors/?q=ai:mastroianni.rita"Locatelli, Ugo"https://zbmath.org/authors/?q=ai:locatelli.ugoSummary: We reconsider the problem of the orbital dynamics of the innermost exoplanet of the \(\upsilon\)-Andromedæsystem (i.e., \(\upsilon\)-And \(\mathbf{b}\)) into the framework of a Secular Quasi-Periodic Restricted Hamiltonian model. This means that we preassign the orbits of the planets that are expected to be the biggest ones in that extrasolar system (namely, \(\upsilon\)-And \(\mathbf{c}\) and \(\upsilon\)-And \(\mathbf{d})\). The Fourier decompositions of their secular motions are injected in the equations describing the orbital dynamics of \(\upsilon\)-And \(\mathbf{b}\) under the gravitational effects exerted by those two exoplanets. By a computer-assisted procedure, we prove the existence of KAM tori corresponding to orbital motions that we consider to be very robust configurations, according to the analysis and the numerical explorations made in our previous article. The computer-assisted assisted proofs are successfully performed for two variants of the Secular Quasi-Periodic Restricted Hamiltonian model, which differs for what concerns the effects of the relativistic corrections on the orbital motion of \(\upsilon\)-And \(\mathbf{b}\), depending on whether they are considered or not.New adiabatic invariants for disturbed non-material volumeshttps://zbmath.org/1532.700232024-05-13T19:39:47.825584Z"Li, Lin"https://zbmath.org/authors/?q=ai:li.lin.12|li.lin.6|li.lin.9|li.lin.2|li.lin.3|li.lin.5|li.lin|li.lin.1The framework of the present paper is the field of non-material volumes, and specifically the dynamics induced by the so-called Mei symmetry for disturbed non-material volumes. After formulating the Lagrangian formalism of such a system and defining the Mei symmetrical perturbation in this context, new adiabatic invariants are obtained which, in particular cases, reproduce the expression of exact invariants previously obtained in the literature. The evolution of these adiabatic invariants is tested on a particular example of application, namely the motion of a rocket under an external force proportional to the displacement times the mass of the rocket.
Reviewer: Fernando Casas (Castellón de la Plana)On isochronicityhttps://zbmath.org/1532.700242024-05-13T19:39:47.825584Z"Treschev, D. V."https://zbmath.org/authors/?q=ai:treshchev.dmitrij-vSummary: We obtain a complete set of explicit necessary and sufficient conditions for the isochronicity of a Hamiltonian system with one degree of freedom. The conditions are presented in terms of the Taylor coefficients of the Hamiltonian function and have the form of an infinite collection of polynomial equations.On auxiliary fields and Lagrangians for relativistic wave equationshttps://zbmath.org/1532.700322024-05-13T19:39:47.825584Z"Sharapov, Alexey"https://zbmath.org/authors/?q=ai:sharapov.alexey-a"Shcherbatov, David"https://zbmath.org/authors/?q=ai:shcherbatov.davidA setup is presented to obtain a Lagrangian formulation for a set of partial differential equations. More specifically, the problem is investigated when the equations are linear partial differential equations with constant coefficients. The setup is used for the case of massive free fields of integer spins.
Reviewer: Mohammad Khorrami (Tehrān)A structure preserving stochastic perturbation of classical water wave theoryhttps://zbmath.org/1532.760262024-05-13T19:39:47.825584Z"Street, Oliver D."https://zbmath.org/authors/?q=ai:street.oliver-dSummary: The inclusion of stochastic terms in equations of motion for fluid problems enables a statistical representation of processes which are left unresolved by numerical computation. Here, we derive stochastic equations for the behaviour of surface gravity waves using an approach which is designed to preserve the geometric structure of the equations of fluid motion beneath the surface. In doing so, we find a stochastic equation for the evolution of a velocity potential and, more significantly, demonstrate that the stochastic equations for water wave dynamics have a Hamiltonian structure which mirrors that found by Zakharov for the deterministic theory. This involves a perturbation of the velocity field which, unlike the deterministic velocity, need not be irrotational for the problem to close.Spacetime geometry of spin, polarization, and wavefunction collapsehttps://zbmath.org/1532.810042024-05-13T19:39:47.825584Z"Beil, Charlie"https://zbmath.org/authors/?q=ai:beil.charlieSummary: To incorporate quantum nonlocality into general relativity, we propose that the preparation and measurement of a quantum system are simultaneous events. To make progress in realizing this proposal, we introduce a spacetime geometry that is endowed with particles which have no distinct points in their worldlines; we call these particles `pointons'. This new geometry recently arose in non-Noetherian algebraic geometry. We show that on such a spacetime, metrics are degenerate and tangent spaces have variable dimension. This variability then implies that pointons are spin-\(\frac{1}{2}\) fermions that satisfy the Born rule, where a projective measurement of spin corresponds to an actual projection of tangent spaces of different dimensions. Furthermore, the 4-velocities of pointons are necessarily replaced by their Hodge duals, and this transfer from vector to pseudo-tensor introduces a free choice of orientation that we identify with electric charge. Finally, a simple composite model of electrons and photons results from the metric degeneracy, and from this we obtain a new ontological model of photon polarization.Finite temperature properties of an integrable zigzag ladder chainhttps://zbmath.org/1532.810402024-05-13T19:39:47.825584Z"Tavares, T. S."https://zbmath.org/authors/?q=ai:tavares.t-sean"Ribeiro, G. A. P."https://zbmath.org/authors/?q=ai:ribeiro.g-a-pSummary: We consider the interaction-round-a-face version of the six-vertex model for arbitrary anisotropy parameter, which allow us to derive an integrable one-dimensional quantum Hamiltonian with three-spin interactions. We apply the quantum transfer matrix approach for the face model version of the six-vertex model. The integrable quantum Hamiltonian shares some thermodynamical properties with the Heisenberg XXZ chain, but has different ordering and critical exponents. Two gapped phases are the dimerized antiferromagnetic order and the usual antiferromagnetic (Néel) order for positive nearest neighbor Ising coupling. In between these, there is an extended critical region, which is a quantum spin-liquid with broken parity symmetry inducing an oscillatory behavior at the long distance \(<\sigma_1^z\sigma_{\ell + 1}^z>\) correlation. At finite temperatures, the numerical solution of the non-linear integral equations allows for the determination of the correlation length as well as for the momentum of the oscillation.Integrable degenerate \(\mathcal{E}\)-models from 4d Chern-Simons theoryhttps://zbmath.org/1532.810422024-05-13T19:39:47.825584Z"Liniado, Joaquin"https://zbmath.org/authors/?q=ai:liniado.joaquin"Vicedo, Benoît"https://zbmath.org/authors/?q=ai:vicedo.benoitSummary: We present a general construction of integrable degenerate \(\mathcal{E}\)-models on a 2d manifold \(\Sigma\) using the formalism of \textit{K. Costello} and \textit{M. Yamazaki} [``Gauge theory and integrability, III'', Preprint, \url{arXiv:1908.02289}] based on 4d Chern-Simons theory on \(\Sigma \times \mathbb{C}P^1\). We begin with a physically motivated review of the mathematical results of \textit{M. Benini} et al. [Commun. Math. Phys. 389, No. 3, 1417--1443 (2022; Zbl 1486.81145)] where a unifying 2d action was obtained from 4d Chern-Simons theory which depends on a pair of 2d fields \(h\) and \(\mathcal{L}\) on \(\Sigma\) subject to a constraint and with \(\mathcal{L}\) depending rationally on the complex coordinate on \(\mathbb{C}P^1\). When the meromorphic 1-form \(\omega\) entering the action of 4d Chern-Simons theory is required to have a double pole at infinity, the constraint between \(h\) and \(\mathcal{L}\) was solved in [\textit{S. Lacroix} and \textit{B. Vicedo}, SIGMA, Symmetry Integrability Geom. Methods Appl. 17, Paper 058, 45 p. (2021; Zbl 1510.17051)] to obtain integrable non-degenerate \(\mathcal{E}\)-models. We extend the latter approach to the most general setting of an arbitrary 1-form \(\omega\) and obtain integrable degenerate \(\mathcal{E}\)-models. To illustrate the procedure, we reproduce two well-known examples of integrable degenerate \(\mathcal{E}\)-models: the pseudo-dual of the principal chiral model and the bi-Yang-Baxter \(\sigma\)-model.Search for the signal of vector-like bottom quark at LHeC in the final state with \(3b\)-jetshttps://zbmath.org/1532.810822024-05-13T19:39:47.825584Z"Zeng, Qing-Guo"https://zbmath.org/authors/?q=ai:zeng.qingguo"Pan, Yu-Si"https://zbmath.org/authors/?q=ai:pan.yu-si"Zhang, Jian"https://zbmath.org/authors/?q=ai:zhang.jian.20|zhang.jian.23|zhang.jian.16|zhang.jian.1|zhang.jian.42|zhang.jian.8|zhang.jian.25|zhang.jian.14|zhang.jian.4|zhang.jian.7|zhang.jian.2Summary: Vector-like quarks arise in many new theories beyond the Standard Model, which is driven by several theoretical issues. In a framework of the Standard Model simply extended by an \(SU(2)\) singlet vector-like bottom quark, we present a study of single production of such vector-like bottom quark at the LHeC. The work is preformed with assuming the LHeC with center of mass energy \(\sqrt{ s} = 1.98\) TeV and an integrated luminosity \(\mathcal{L} = 1000\mathrm{fb}^{-1}\). We focus on the process \(e^- p \to \nu_e B\to \nu_e b H(\mathrm{orZ}) \to\nu_{\mathrm{e}} \mathrm{bb}\overline{\mathrm{b}}\). The signal is searched in events with three \(b\)-jets within final state. Exclusion (discover) capability of the LHeC for the signal of the vector-like bottom quark are obtained at \(2\sigma\) (\(5\sigma\)) confidence level. Our result demonstrates a significant improvement compared to previous relevant researches. We also discuss the capability of LHeC for the singlet \(B\) search with combined analysis from other experiments.Structure of the plane waves for a spin 3/2 particlehttps://zbmath.org/1532.810942024-05-13T19:39:47.825584Z"Ivashkevich, A. V."https://zbmath.org/authors/?q=ai:ivashkevich.alina-vSummary: The structure of the plane waves solutions for a relativistic spin 3/2 particle described by 16-component vector-bispinor is studied. In massless case, two representations are used: Rarita - Schwinger basis, and a special second basis in which the wave equation contains the Levi-Civita tensor. In the second representation it becomes evident the existence of gauge solutions in the form of 4-gradient of an arbitrary bispinor. General solution of the massless equation consists of six independent components, it is proved in an explicit form that four of them may be identified with the gauge solutions, and therefore may be removed. This procedure is performed in the Rarita - Schwinger basis as well. For the massive case, in Rarita - Schwinger basis four independent solutions are constructed explicitly.