Computing traveling wave solutions of the fifth order Korteweg-de Vries equation through Whitham theory
Whitham modulation theory is a useful tool for describing the adiabatic evolution of periodic solutions to nonlinear dispersive equations. Modulation theory yields a first order system of quasilinear partial differential equations which describe the evolution of the wave parameters. Whitham theory is particularly useful to construct dispersive shock waves (DSWs) which are classically represented by a continuous rarefaction wave solution of the modulation equations.
In this talk we discuss discontinuous shock solutions to the fifth order Korteweg-de Vries Whitham equations. These shocks are used to construct new types of traveling wave solutions which connect disparate far-field periodic waves and satisfy the Rankine-Hugoniot jump conditions. The results presented here can be extended to construct traveling wave solutions to many systems exhibiting high order dispersion. We will conclude this presentation by discussing an experimental setup at CU where these traveling waves are observed.