Condensed Matter Physics, 2010, vol. 13, No. 4, p. 43002:1-21
A generalized hydrodynamical Gurevich-Zybin equation of Riemann type and its Lax type integrability
(Department of Mathematical Physics, Lebedev Physics Institute of RAS, Moscow, Russian Federation)
(The AGH University of Science and Technology, Krakow, 30-059, Poland; The Ivan Franko State Pedagogical University, Drohobych of Lviv region, Ukraine)
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.
Riemann type hydrodynamical equations, Lax type integrability, conservation laws