The gradient-holonomic integrability analysis of a Whitham-type nonlinear dynamical model for a relaxing medium with spatial memory

Abstract

A new Whitham-type nonlinear evolution equation describing short-wave perturbations in a relaxing medium is studied. Making use of the gradientholonomic analysis, the bi-Hamiltonicity and complete integrability of the corresponding dynamical system is stated. An infinite hierarchy of dispersive conservation laws which commute with each other is constructed. The twoand four-dimensional invariant reductions are studied in detail.

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

“…Moreover, if we further apply the reduction η = 0, we obtain, respectively, new non-polynomial conservation laws for the Hunter-Saxton dynamical system (27), supplementing those found before in [22,24].…”

“…The corresponding recursion operator proves to generate no new conservation law, if one applies it to the non-polynomial conservations laws (24).…”

“…This equation allows a simple reduction to the Hunter-Saxton dynamical system [13,16,21,22,24] at η = 0 :…”

“…The non-polynomial conservation laws (24), upon rewriting with respect to the substitution (25), give rise to the related non-polynomial conservation laws for a generalized two-component HunterSaxton dynamical system (27). Moreover, if we further apply the reduction η = 0, we obtain, respectively, new non-polynomial conservation laws for the Hunter-Saxton dynamical system (27), supplementing those found before in [22,24].…”

A vertex operator representation of solutions to the Gurevich-Zybin hydrodynamical equation

“…These methods appeared to be very effective [1] in investigating many types of nonlinear spatially one-dimensional systems of hydrodynamical type and, in particular, the characteristics method in the form of a "reciprocal" transformation of variables has been used recently in studying a so called Gurevich-Zybin system [2,3] in [9] and a Whitham type system in [5,6]. Moreover, this method was further effectively applied to studying solutions to a generalized [5] where N ∈ Z + , u ∈ M 1 ⊂ C ∞ (R/2πZ; R) is a smooth function on a periodic functional manifold M 1 and t ∈ R is the evolution parameter.…”