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

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.

References in corpus (6)

• Dispersive shock waves and modulation theory
• Resolution of a shock in hyperbolic systems modified by weak dispersion
• Undular bore theory for the Gardner equation
• Breaking of Galilean invariance in the hydrodynamic formulation of ferromagnetic thin films
• Analytic model for a frictional shallow-water undular bore
• Dispersive hydrodynamics of nonlinear polarization waves in two-component Bose-Einstein condensates

Cited by in corpus (5)

• Domain wall problem in the quantum XXZ chain and semiclassical behavior close to the isotropic point
• Solution of the Riemann problem for polarization waves in a two-component Bose-Einstein condensate
• Riemann problem for the light pulses in optical fibers for the generalized Chen-Lee-Liu equation
• Riemann problem for the photon fluid: self-steepening effects
• Trigonometric shock waves for the Kaup-Boussinesq system