Trigonometric shock waves in the Kaup–Boussinesq system

Sergey K Ivanov,Anatoly M Kamchatnov

doi:10.1007/s11071-022-07326-5

Abstract

We consider the modulationally stable version of the Kaup-Boussinesq system which models propagation of nonlinear waves in various physical systems. It is shown that the Whitham modulation equations for this model have a new type of solutions which describe trigonometric shock waves. In the Gurevich-Pitaevskii problem of evolution of an initial discontinuity, these solutions correspond to a non-zero wave excitation on one of the sides of the discontinuity. Our analytical results are confirmed by numerical calculations.

Highlights

Dispersion in nonlinear systems can dramatically affect wave profile leading to a host of new physical wave structures such as solitons and dispersive shock waves
Dispersive shock waves, called undular bores in fluid mechanics applications, are represented as modulated nonlinear periodic waves and the process of their formation and evolution is described in the Gurevich-Pitaevskii approximation [1] by the Whitham theory of modulations [2]
The original formulation of Gurevich and Pitaevskii approach was applied to description of expanding collisionless shocks in framework of the Whitham-averaged equations for the integrable Korteweg-de Vries (KdV) equation [5, 6]

Summary

INTRODUCTION

Dispersion in nonlinear systems can dramatically affect wave profile leading to a host of new physical wave structures such as solitons and dispersive shock waves. Dispersive shock waves, called undular bores in fluid mechanics applications, are represented as modulated nonlinear periodic waves and the process of their formation and evolution is described in the Gurevich-Pitaevskii approximation [1] by the Whitham theory of modulations [2] (for reviews see [3, 4]). We consider the initial states of a different type: we assume that on one side of the initial discontinuity the profiles are represented by periodic solutions rather than by uniform distributions, as it supposed in the standard Riemann problem. This means that our theory describes spreading out of the front of the nonlinear wave excitation along the rarefaction wave. This opens a new route to analytical description of wave structures arising from more complex initial states

PERIODIC SOLUTIONS AND WHITHAM EQUATIONS

Periodic solutions

Whitham modulation equations

TRIGONOMETRIC SHOCK WAVE

CONCLUSION