A generalized hydrodynamical Gurevich-Zybin equation of Riemann type and its Lax type integrability

Abstract

This paper is devoted to the study of a hydrodynamical equation of Riemann type, generalizing the remarkable Gurevich-Zybin system. This multi-component non-homogenous hydrodynamic equation is characterized by the only characteristic flow velocity. The compatible bi-Hamiltonian structures and Lax type representations of the 3-and 4-component generalized Riemann type hydrodynamical system are analyzed. For the first time the obtained results augment the theory of integrability of hydrodynamic type systems, originally developed only for distinct characteristic velocities in homogenous case.

Quasi-linearization and stability analysis of some self-dual, dark equations and a new dynamical system †

“…tively, and which prove to be completely integrable bi-Hamiltonian systems, actively studied [5,[17][18][19][20][21][22] during past decades. Some other more nontrivial examples can be taken from the work [4], devoted to description of the integrable hierarchies of modified Burgers type nonlinear dynamical systems.…”