Expansion shock waves in regularised shallow water theory

TL;DR

This paper discovers a new type of stationary expansion shock wave in regularised shallow water equations, showing its unique properties, persistence, and robustness through analytical and numerical methods.

Contribution

It introduces the concept of expansion shocks in regularised shallow water models and analyzes their formation, persistence, and decay mechanisms.

Findings

Expansion shocks violate classical entropy conditions.

Expansion shocks persist and decay algebraically over time.

Robustness of expansion shocks under weak dissipation and asymmetric conditions.

Abstract

We identify a new type of shock wave by constructing a stationary expansion shock solution of a class of regularised shallow water equations that include the Benjamin-Bona-Mahoney (BBM) and Boussinesq equations. An expansion shock exhibits divergent characteristics, thereby contravening the classical Lax entropy condition. The persistence of the expansion shock in initial value problems is analysed and justified using matched asymptotic expansions and numerical simulations. The expansion shock's existence is traced to the presence of a non-local dispersive term in the governing equation. We establish the algebraic decay of the shock as it is gradually eroded by a simple wave on either side. More generally, we observe a robustness of the expansion shock in the presence of weak dissipation and in simulations of asymmetric initial conditions where a train of solitary waves is shed from one side of the shock.