Stability and dynamical features of solitary wave solutions for a hydrodynamic-type system taking into account non-local effects

TL;DR

This paper analyzes a hydrodynamic system with nonlocal effects, revealing the existence of solitary wave solutions, their stability properties, and their interactions through symmetry analysis and numerical simulations.

Contribution

It introduces a new hydrodynamic model incorporating nonlocal effects and studies the stability and interaction of its solitary wave solutions.

Findings

Homoclinic solutions describe solitary waves of compression or rarefication.

Waves of compression are unstable, while waves of rarefication are stable.

Rarefaction solitary waves maintain shape after interaction.

Abstract

We consider a hydrodynamic-type system of balance equations which is closed by the dynamic equation of state taking into account the effects of spatial nonlocality. Symmetry and local conservation laws of this system are studied. A system of ODEs being obtained via the group theory reduction of the initial system of PDEs is investigated. The reduced system is shown to possess a family of the homoclinic solutions. Depending on the values of the parameters, the homoclinic solutions describe the solitary waves of compression or rarefication. The solutions corresponding to the wave of compression are shown to be unstable. More likely, the waves of rarefication are stable. Numerical simulations demonstrate that the solitary waves of rarefication moving toward each other maintain their shape after the interaction.