On the integrability of a new generalized Gurevich—Zybin dynamical system, its Hunter—Saxton type reduction and related mysterious symmetries

Abstract

There is studied the integrability of a generalized Gurevich-Zybin dynamical system based on the differential-algebraic and geometrically motivated gradient-holonomic approaches. There is constructed the corresponding Lax type represenation, compatible Poisson structures as well as the integrability of the related Hunter-Saxton reduction. In particular, there are constructed its Lax type repreentation, the Hamiltonian symmetries as flows on a functional manifold endowed with compatible Poisson structures as well as so called new mysterious symmetries, depending on functional parameter. Similar results are also presented for the potential-KdVdynamical system, for which we also obtained its new mysterious symmetries first presented in a clear, enough short and analytically readable form.

Keywords

Bibliographic record

BibTeX Citation

@article{Blackmore2022integrabilitygeneralized,
  author = {Blackmore, D. and Prykarpatsky, Ya. and Prytula, M. and Dutykh, D. and Prykarpatsky, A.},
  title = {On the integrability of a new generalized Gurevich—Zybin dynamical system, its Hunter—Saxton type reduction and related mysterious symmetries},
  journal = {Analysis and Mathematical Physics},
  year = {2022},
  volume = {12},
  number = {66},
  pages = {66--66},
  doi = {10.1007/s13324-022-00662-0},
  abstract = {There is studied the integrability of a generalized Gurevich-Zybin dynamical system based on the differential-algebraic and geometrically motivated gradient-holonomic approaches. There is constructed the corresponding Lax type represenation, compatible Poisson structures as well as the integrability of the related Hunter-Saxton reduction. In particular, there are constructed its Lax type repreentation, the Hamiltonian symmetries as flows on a functional manifold endowed with compatible Poisson structures as well as so called new mysterious symmetries, depending on functional parameter. Similar results are also presented for the potential-KdVdynamical system, for which we also obtained its new mysterious symmetries first presented in a clear, enough short and analytically readable form.},
  keywords = {37K35, Differential-algebraic analysis, Reduced Hunter-Saxton dynamical system, Potential-Korteweg-de Vries dynamical system, Generalized Gurevich-Zybin dynamical system, Compatible Poisson structures, 35G25, Asymptotic analysis, 35Q53, 58J70, Integrability, Symmetry analysis, 35N10, 34A34, Hamiltonian system, 17B68, 37K10, 58J72, Mysterious symmetries, 37K05, Conservation laws, 17B80}
}