Nonlinear Schr\"odinger equations and the universal description of dispersive shock wave structure

TL;DR

This paper develops a universal analytical framework for describing dispersive shock waves in nonlinear dispersive equations, extending existing theories and validated through numerical simulations.

Contribution

It introduces a method that unifies NLS and Whitham modulation equations to analyze DSW structure across various equations, including non-integrable cases.

Findings

Accurate DSW structure description from harmonic edge to interior.

Extension of DSW fitting theory to include interior structure.

Validation through numerical simulations with higher order NLS terms.

Abstract

The nonlinear Schr\"odinger (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 non-integrable nonlinear dispersive equations. The proposed method extends DSW fitting theory that prescribes the motion of a DSW's edges into the DSW's interior, i.e., 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.