Title: Nonlinear wave modulations in systems with higher order dispersion

Abstract:Scalar equations modeling nonlinear wave evolution typically take the form of a conservation law often modified by long wave dispersion. However, some features of the original problem may be lost when higher order dispersive effects are neglected. One may modify the model equation and include full linear dispersion by use of a conservation law or higher order long wave dispersion by including more differential terms. In this presentation, we derive the set of Whitham modulation equations for a general class of nonlinear, dispersive equations with general nonlinearity and full linear dispersion. The Whitham modulation equations describe the slow evolution of a nonlinear, periodic wave. Following the derivation of the Whitham modulation equations, we use the fifth order Korteweg-de Vries equation as an example to demonstrate the novel nonlinear phenomena that can be described by the modulation system when dispersive terms higher than third order are included. In this portion of the talk, we focus on traveling wave solutions that correspond to discontinuous shock solutions of the Whitham modulation equations.