Stationary Expansion Shocks for a Regularized Boussinesq System

TL;DR

This paper extends the analysis of stationary expansion shocks from a scalar equation to a Boussinesq system, using advanced asymptotic methods and confirming results with numerical simulations.

Contribution

It develops a novel asymptotic approach for stationary expansion shocks in the Boussinesq system, incorporating Riemann invariants and high-order matched asymptotics.

Findings

Asymptotic solutions agree well with numerical simulations

Extension from scalar to system case is successfully achieved

Method provides insights into dispersive shock structures in shallow water

Abstract

Stationary expansion shocks have been recently identified as a new type of solution to hyperbolic conservation laws regularized by non-local dispersive terms that naturally arise in shallow-water theory. These expansion shocks were studied in (El, Hoefer, Shearer 2016) for the Benjamin-Bona-Mahony equation using matched asymptotic expansions. In this paper, we extend the analysis of (El, Hoefer, Shearer 2016) to the regularized Boussinesq system by using Riemann invariants of the underlying dispersionless shallow water equations. The extension for a system is non-trivial, requiring a combination of small amplitude, long-wave expansions with high order matched asymptotics. The constructed asymptotic solution is shown to be in excellent agreement with accurate numerical simulations of the Boussinesq system for a range of appropriately smoothed Riemann data.