Expansion shock waves in regularised shallow water theory

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.

6 pages, 5 figures

References in corpus (1)

• Dispersive shock waves and modulation theory

Cited by in corpus (5)

• Soliton gas in bidirectional dispersive hydrodynamics
• Dispersive Riemann problem for the Benjamin-Bona-Mahony equation
• Rankine--Hugoniot conditions for fluids whose energy depends on space and time derivatives of density
• Asymptotic soliton like solutions to the singularly perturbed Benjamin-Bona-Mahony equation with variable coefficients
• Stationary Expansion Shocks for a Regularized Boussinesq System