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

Abstract

Abstract Whitham modulation theory describes the zero dispersion limit of nonlinear disperesive partial differential equations (PDEs) by a system of conservation laws for the parameters of modulated periodic traveling waves (TWs). In this work, admissible, discontinuous, weak solutions of the Whitham modulation equations—termed Whitham shocks —are identified with zero dispersion limits of TW solutions to higher order dispersive PDEs. The far-field behavior of the TW solutions satisfies the Rankine–Hugoniot jump conditions for the Whitham modulation equations. 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 where three admissible one-parameter families of Whitham shocks are identified as solution components to the generalized Riemann problem for the Whitham modulation equations. Admissible KdV5–Whitham shocks are generally undercompressive, i.e., all characteristic families pass through the shock. The heteroclinic TWs 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 TW solutions to dispersive nonlinear wave equations and a framework to identify physically relevant, admissible shock solutions of the Whitham modulation equations.

Topics & Concepts