Nonlinear Schrödinger equations and the universal description of dispersive shock wave structure

T Congy,M.A Hoefer,M Shearer,G.A El

doi:10.1111/sapm.12247

Abstract

AbstractThe nonlinear Schrödinger (NLS) equation and the Whitham modulation equations both describe slowly varying, locally periodic nonlinear wavetrains, albeit in differing amplitude‐frequency domains. In this paper, we take advantage of the overlapping asymptotic regime that applies to both the NLS and Whitham modulation descriptions in order to develop a universal analytical description of dispersive shock waves (DSWs) generated in Riemann problems for a broad class of integrable and nonintegrable nonlinear dispersive equations. The proposed method extends DSW fitting theory that prescribes the motion of a DSW's edges into the DSW's interior, that is, this work reveals the DSW structure. Our approach also provides a natural framework in which to analyze DSW stability. We consider several representative, physically relevant examples that illustrate the efficacy of the developed general theory. Comparisons with direct numerical simulations show that inclusion of higher order terms in the NLS equation enables a remarkably accurate description of the DSW structure in a broad region that extends from the harmonic, small amplitude edge.

Highlights

There has been a surge of interest recently in the subject of dispersive hydrodynamics and, in particular, dispersive shock waves (DSWs)
The key element of the extension is the realization that the DSW modulation described by an expansion fan solution of the Whitham modulation equations can be universally approximated, in the vicinity of the weakly nonlinear harmonic edge, by a special vacuum rarefaction solution of the shallow water equations
The connection between the original dispersive hydrodynamics and the approximating shallow water system occurs via a long-wave, dispersionless limit of the nonlinear Schrodinger (NLS) equation for weakly nonlinear, narrow-band Stokes waves, whose parameters are determined by DSW fitting when the NLS equation is of defocusing type

Summary

Introduction

There has been a surge of interest recently in the subject of dispersive hydrodynamics and, in particular, dispersive shock waves (DSWs) (see [7, 19] and references therein). We develop a general method for the determination of the universal nonlinear DSW modulation—the DSW structure—near the harmonic edge This asymptotic modulation provides crucial information about the variation of the amplitude in the DSW (i.e. the envelope of the oscillatory wavetrain) as well as other physical DSW parameters such as the wavenumber and mean flow. This is used to derive the universal, first order approximation of the DSW modulation as a vacuum rarefaction simple wave solution of the long-wave, dispersionless limit of the NLS equation. Appendix C is devoted to a brief description of numerical methods used for simulations

Dispersive shock waves: modulation theory

The determination of the harmonic edge

Small amplitude DSW regime and the NLS equation

Higher order NLS approximation

NLS description of dispersive shock waves

Conclusion and Discussion