Shock waves of spherical/cylindrical KdV-B: Asymptotic, stability, superposition

TL;DR

This paper analyzes shock wave solutions of spherical and cylindrical KdV-B equations, focusing on their asymptotic behavior, stability, and superposition rules, with implications for a broad class of shock waves.

Contribution

It provides a detailed asymptotic analysis and stability proof for shock wave solutions, establishing effective superposition rules applicable to various shock wave types.

Findings

Derived asymptotic descriptions of shock wave solutions.

Proved stability of these shock solutions using conservation laws.

Established superposition rules applicable to discontinuous shock waves.

Abstract

Spherical and cylindrical KdV-B equations have few known exact solutions, yet these solutions are hard to be interpreted physically. But these equations do have a family of diverging shock waves. Their properties such as asymptotic modes, stability, rules of their interactions/superposition are the subject of this paper. It gives a detailed asymptotic description of the one-parameter families of shock wave solutions and proves their stability using a conservation law. Based on these results, effective rules of superposition are obtained. Moreover these rules are applicable to a wide class of shock waves, in particular discontinuous. Typical examples are illustrated by graphs.