Dispersive shock waves theory for non-integrable equations

TL;DR

This paper develops a method based on Whitham modulation theory to calculate dispersive shock wave parameters in non-integrable equations, validated through comparisons with integrable cases and numerical simulations.

Contribution

It introduces a universal approach for dispersive shock wave analysis applicable to non-integrable equations using Whitham's conservation laws.

Findings

Explicit edge motion laws derived

Method validated with KdV and NLS equations

Good agreement with numerical simulations

Abstract

We suggest a method for calculation of parameters of dispersive shock waves in framework of Whitham modulation theory applied to non-integrable wave equations with a wide class of initial conditions corresponding to propagation of a pulse into a medium at rest. The method is based on universal applicability of Whitham's `number of waves conservation law' as well as on the conjecture of applicability of its soliton counterpart to the above mentioned class of initial conditions which is substantiated by comparison with similar situations in the case of completely integrable wave equations. This allows one to calculate the limiting characteristic velocities of the Whitham modulation equations at the boundary with the smooth part of the pulse whose evolution obeys the dispersionless approximation equations. We show that explicit analytic expressions can be obtained for laws of motion of the edges. The validity of the method is confirmed by its application to similar situations described by the integrable Korteweg-de Vries (KdV) and nonlinear Schr\"{o}dinger (NLS) equations and by comparison with the results of numerical simulations for the generalized KdV and NLS equations.