Recent zbMATH articles in MSC 37J35https://www.zbmath.org/atom/cc/37J352022-04-28T18:15:44.554837ZWerkzeugGeneralized analytic integrability of a class of polynomial differential systems in \(\mathbb{C}^2\)https://www.zbmath.org/1482.370272022-04-28T18:15:44.554837Z"Llibre, Jaume"https://www.zbmath.org/authors/?q=ai:llibre.jaume"Tian, Yuzhou"https://www.zbmath.org/authors/?q=ai:tian.yuzhouThe paper is focused on two-dimensional autonomous complex polynomial ordinary differential systems with separable variables. Necessary and sufficient conditions for the existence of a first autonomous integral in the classes of generalized analytic and polynomial functions are obtained. Some examples are given and discussed.
Reviewer: Valentine Tyshchenko (Grodno)On the existence of focus singularities in one model of a Lagrange top with a vibrating suspension pointhttps://www.zbmath.org/1482.370542022-04-28T18:15:44.554837Z"Borisov, A. V."https://www.zbmath.org/authors/?q=ai:borisov.alexey-v"Ryabov, P. E."https://www.zbmath.org/authors/?q=ai:ryabov.p-e"Sokolov, S. V."https://www.zbmath.org/authors/?q=ai:sokolov.sergei-vSummary: We consider a completely integrable Hamiltonian system with two degrees of freedom that describes the dynamics of a Lagrange top with a vibrating suspension point. The results of a stability analysis of equilibrium positions are clearly presented. It turns out that, in the case of a vibrating suspension point, both equilibrium positions can be unstable, which corresponds to the existence of focus singularities in the considered model.Billiards with changing geometry and their connection with the implementation of the Zhukovsky and Kovalevskaya caseshttps://www.zbmath.org/1482.370552022-04-28T18:15:44.554837Z"Fomenko, A. T."https://www.zbmath.org/authors/?q=ai:fomenko.anatolii-t"Vedyushkina, V. V."https://www.zbmath.org/authors/?q=ai:vedyushkina.viktoriya-viktorovnaSummary: The paper presents a class of billiards with varying geometry, the so-called force or evolutionary billiards, which enable us to realize, in the sense of Liouville equivalence, the well-known cases of Zhukovsky and Kovalevskaya for certain energy zones. On the corresponding 4-dimensional open phase submanifolds, the indicated systems are implemented for an increase in energy on all successively occurring isoenergy 3-surfaces.Integrability of point-vortex dynamics via symplectic reduction: a surveyhttps://www.zbmath.org/1482.370562022-04-28T18:15:44.554837Z"Modin, Klas"https://www.zbmath.org/authors/?q=ai:modin.klas"Viviani, Milo"https://www.zbmath.org/authors/?q=ai:viviani.miloThe authors consider point-vertex dynamics and their integrability. This dynamics is described via idealized non-smooth solutions to the incompressible Euler equations on two-dimensional manifolds. The aim is to provide a unified treatment for proving integrability results for 2-, 3-, or 4-point-vertices. Part of their goal is to show how the symplectic reduction can provide a broader approach for proving integrability results, especially for point-vertex dynamics.
Euler equations on an orientable Riemannian manifold that govern an incompressible inviscid fluid have the form \(\dot{\mathbf{v}} + \nabla _\mathbf{v} \mathbf{v} = - \nabla p\) with div \(\mathbf{v} = 0\), where \(\mathbf{v}\) is a vector field on a manifold \(M\) incorporating the motion of the fluid's particles, \(p\) is the pressure function and \(\nabla _\mathbf{v}\) is the covariant derivative along \(\mathbf{v}\). \textit{H. Helmholtz} [J. Reine Angew. Math. 55, 25--55 (1858; ERAM 055.1448cj)] showed that the two-dimensional Euler equations have special solutions with a finite number of point-vertices. These solutions are not smooth and are characterized by the vorticity \mbox{curl \(\mathbf{v} = \sum_{i=1} ^ {n} \Gamma _i \delta_{\mathbf{r}_i}\)} where non-zero \(\Gamma_i\) is the strength of the vortex \(i\), \(\mathbf{r}_i\) is its position, and \(\delta_{\mathbf{r}_i}\) is a delta function.
The authors describe point-vertex equations and their Hamiltonian structures on the sphere, the plane, the hyperbolic plane, and the flat torus. Each of these cases has a different symmetry group, and the symplectic reduction is treated separately in each case to establish integrability.
The authors also briefly review nonintegrability results.
They conclude with some observations about how their results pertain to long term predictions for the Euler equations. An appendix offers a kind of visual portrait of point-vortex solutions.
Reviewer: William J. Satzer Jr. (St. Paul)New cases of homogeneous integrable systems with dissipation on tangent bundles of two-dimensional manifoldshttps://www.zbmath.org/1482.370572022-04-28T18:15:44.554837Z"Shamolin, M. V."https://www.zbmath.org/authors/?q=ai:shamolin.m-vSummary: The integrability of certain classes of homogeneous dynamical systems on the tangent bundles of two-dimensional manifolds is shown. The force fields involved in the systems lead to dissipation of variable sign and generalize previously considered fields.New cases of homogeneous integrable systems with dissipation on tangent bundles of three-dimensional manifoldshttps://www.zbmath.org/1482.370582022-04-28T18:15:44.554837Z"Shamolin, M. V."https://www.zbmath.org/authors/?q=ai:shamolin.m-vSummary: The integrability of certain classes of homogeneous dynamical systems on the tangent bundles of three-dimensional manifolds is shown. The force fields involved in the systems lead to dissipation of variable sign and generalize previously considered fields.New cases of homogeneous integrable systems with dissipation on tangent bundles of four-dimensional manifoldshttps://www.zbmath.org/1482.370592022-04-28T18:15:44.554837Z"Shamolin, M. V."https://www.zbmath.org/authors/?q=ai:shamolin.m-vSummary: The integrability of certain classes of homogeneous dynamical systems on the tangent bundles of four-dimensional manifolds is shown. The force fields involved in the systems lead to dissipation of variable sign and generalize previously considered fields.Force evolutionary billiards and billiard equivalence of the Euler and Lagrange caseshttps://www.zbmath.org/1482.370602022-04-28T18:15:44.554837Z"Vedyushkina, V. V."https://www.zbmath.org/authors/?q=ai:vedyushkina.viktoriya-viktorovna"Fomenko, A. T."https://www.zbmath.org/authors/?q=ai:fomenko.anatolii-tSummary: A class of force evolutionary billiards is discovered that realizes important integrable Hamiltonian systems on all regular isoenergy 3-surfaces simultaneously, i.e., on the phase 4-space. It is proved that the well-known Euler and Lagrange integrable systems are billiard equivalent, although the degrees of their integrals are different (two and one).Notes about the KP/BKP correspondencehttps://www.zbmath.org/1482.810132022-04-28T18:15:44.554837Z"Orlov, A. Yu."https://www.zbmath.org/authors/?q=ai:orlov.aleksandr-yuSummary: We present a set of remarks related to previous work. These are remarks on polynomials solutions, the application of the Wick theorem, examples of creation of polynomial solutions with the help of vertex operators, the eigenproblem for polynomials, and a remark on the conjecture by Alexandrov and Mironov, Morozov about the ratios of the projective Schur functions. New results on the bilinear relations between characters of the symmetric group and the Sergeev group and on bilinear relations between skew Schur and projective Schur functions and also between shifted Schur and projective Schur functions are added. Certain new matrix models are discussed.