On the spectral stability of soliton-like solutions to a non-local hydrodynamic-type model

TL;DR

This paper investigates the spectral stability of soliton-like solutions in a nonlinear PDE model of elastic media with internal structures, demonstrating their stability under specific parameter conditions.

Contribution

It introduces a non-local hydrodynamic-type model for elastic media with internal structures and analyzes the spectral stability of its soliton-like solutions.

Findings

Soliton-like solutions exist in the model.

These solutions are spectrally stable under certain parameter restrictions.

The stability analysis provides conditions for wave propagation in complex media.

Abstract

A model of nonlinear elastic medium with internal structure is considered. The medium is assumed to contain cavities, microcracks or blotches of substances that differ sharply in physical properties from the base material. To describe the wave processes in such a medium, the averaged values of physical fields are used. This leads to nonlinear evolutionary PDEs, differing from the classical balance equations. The system under consideration possesses a family of invariant soliton-like solutions. These solutions are shown to be spectrally stable under certain restrictions on the parameters.