Analytical Properties of an Ostrovsky-Whitham Type Dynamical System for a Relaxing Medium with Spatial Memory and its Integrable Regularization

TL;DR

This paper investigates an Ostrovsky-Whitham type dynamical system modeling relaxing media with spatial memory, establishing its integrability, bi-Hamiltonian structure, and deriving an infinite hierarchy of conservation laws.

Contribution

It introduces a new integrable regularization of the system and analyzes its invariant reductions, expanding understanding of its mathematical properties.

Findings

Proves bi-Hamiltonicity and integrability of the system

Derives an infinite hierarchy of conservation laws

Constructs a well-defined regularization with Lax integrability

Abstract

Short-wave perturbations in a relaxing medium, governed by a special reduction of the Ostrovsky evolution equation, and later derived by Whitham, are studied using the gradient-holonomic integrability algorithm.The bi-Hamiltonicity and complete integrability of the corresponding dynamical system is stated and an infinite hierarchy of commuting to each other conservation laws of dispersive type are found. The two- and four-dimensional invariant reductions are studied in detail. The well defined regularization of the model is constructed and its Lax type integrability is discussed.