Evolution of initial discontinuities in the Riemann problem for the Kaup-Boussinesq equation with positive dispersion

TL;DR

This paper analyzes the evolution of initial discontinuities in a positive dispersion Kaup-Boussinesq model, classifying wave structures and confirming results through numerical simulations.

Contribution

It provides a comprehensive classification of wave configurations for the Kaup-Boussinesq system with positive dispersion, including analytical and numerical analysis.

Findings

Classification of all possible wave configurations

Derivation of Whitham modulation equations

Confirmation of analytical results with numerical simulations

Abstract

We consider the space-time evolution of initial discontinuities of depth and flow velocity for an integrable version of the shallow water Boussinesq system introduced by Kaup. We focus on a specific version of this "Kaup-Boussinesq model" for which a flat water surface is modulationally stable, we speak below of "positive dispersion" model. This model also appears as an approximation to the equations governing the dynamics of polarisation waves in two-component Bose-Einstein condensates. We describe its periodic solutions and the corresponding Whitham modulation equations. The self-similar, one-phase wave structures are composed of different building blocks which are studied in detail. This makes it possible to establish a classification of all the possible wave configurations evolving from initial discontinuities. The analytic results are confirmed by numerical simulations.