Superposition of shock waves of the generalized BBM equation

TL;DR

This paper investigates the superposition and interaction rules of shock wave solutions in a generalized BBM equation with dissipation, establishing their stability and broader applicability beyond classical solitons.

Contribution

It provides a detailed analysis of shock wave solutions in the generalized BBM equation, proving their stability and deriving effective superposition rules applicable to various shock waves.

Findings

Shock wave solutions form a two-parameter family.

Stability of solutions is proven using conservation laws.

Superposition rules are applicable to a wide class of shock waves.

Abstract

The generalized BBM studied in this paper contains an additional dissipative term. Thus instead of solitons for the classic BBM there exists a lot of travelling shock wave solutions. The rules of their interactions or superposition is of high importance. The paper gives a detailed description of the two-parameter family of travelling 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 not exclusively to the travelling wave solutions of BBM, but also to a wider class of shock waves, in particular discontinuous. Characteristic examples are illustrated by numerically worked out graphs.