Discontinuous shock solutions of the Whitham modulation equations as dispersionless limits of traveling waves

Abstract

Whitham modulation theory describes the zero dispersion limit of nonlinear waves by a system of conservation laws for the parameters of modulated periodic traveling waves. Here, admissible, discontinuous, weak solutions of the Whitham modulation equations--termed Whitham shocks--are identified with zero dispersion limits of traveling wave solutions to higher order dispersive partial differential equations (PDEs). The far-field behavior of the traveling wave solutions satisfies the Rankine-Hugoniot jump conditions. Generally, the numerically computed traveling waves represent heteroclinic connections between two periodic orbits of an ordinary differential equation. The focus here is on the fifth order Korteweg-de Vries equation. The three admissible one-parameter families of Whitham shocks are used as solution components for the generalized Riemann problem of the Whitham modulation equations. Admissible KdV5-Whitham shocks are generally undercompressive, i.e., all characteristic families pass through the shock. The heteroclinic traveling waves that limit to admissible Whitham shocks are found to be ubiquitous in numerical simulations of smoothed step initial conditions for other higher order dispersive equations including the Kawahara equation (with third and fifth order dispersion) and a nonlocal model of weakly nonlinear gravity-capillary waves with full dispersion. Whitham shocks are linked to recent studies of nonlinear higher order dispersive waves in optics and ultracold atomic gases. The approach presented here provides a novel method for constructing new traveling wave solutions to dispersive nonlinear wave equations and a framework to identify physically relevant, admissible shock solutions of the Whitham modulation equations.