Dispersive shock waves, traveling fronts, and defects in gravity capillary water waves

Presented by Patrick Sprenger (Cambridge, UK)

Dispersive shock waves (DSWs) are multiscale, nonlinear waves that regularize singularities in dispersive hydrodynamic systems. A common mathematical description DSWs is via a self-similar rarefaction wave solution of the Whitham modulation equations. In shallow water waves with sufficiently strong surface tension, numerical simulation of the Kawahara equation reveals a coherent, nonlinear wave structure quite different from the typical DSW.

In certain regimes, this DSW is partially described in terms of a discontinuous shock solution of the Whitham modulation equations that satisfies Rankine-Hugoniot jump conditions. Further analysis of the jump conditions reveals families of traveling wave solutions of the Kawahara equation that asymptote to distinct periodic orbits at infinity.  Each branch terminates at an equilibrium-to-periodic solution in which the equilibrium is the background for a solitary wave that connects to the associated periodic solution.

This family of traveling wave solutions are used to compute traveling defect solutions, which are localized on a periodic background. Extensions of this work to Boussinesq systems modeling gravity-capillary water waves will also be discussed.