  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.
