Abstract

The nonlinear nonlinear Schrödinger (NLS) equation and the Whitham modulation equations both describe slowly varying, locally periodic nonlinear wavetrains, albeit in differing amplitude-frequency domains. Taking advantage of the overlap regime for the applicability of the NLS equation and the Whitham modulation theory, we 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 and complements the DSW fitting theory prescribing the motion of DSW edges. We consider several representative physically relevant examples illustrating 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 modulation in a broad vicinity of the harmonic, small amplitude, edge.This is joint work with Gennady El, Mark Hoefer and Michael Shearer.