On the Whitham Equations for the Defocusing Complex Modified KdV Equation

TL;DR

This paper investigates the Whitham equations for the defocusing complex mKdV equation, revealing unique dispersive shock structures due to their weak hyperbolicity, contrasting with the strictly hyperbolic NLS case.

Contribution

It analyzes the properties of the Whitham equations for the defocusing complex mKdV, highlighting their weak hyperbolicity and resulting novel dispersive shock structures.

Findings

Whitham equations for mKdV are not strictly hyperbolic.

Weak hyperbolicity leads to unique dispersive shock structures.

Comparison with NLS shows different hyperbolic properties and shock behaviors.

Abstract

We study the Whitham equations for the defocusing complex modified KdV (mKdV) equation. These Whitham equations are quasilinear hyperbolic equations and they describe the averaged dynamics of the rapid oscillations which appear in the solution of the mKdV equation when the dispersive parameter is small. The oscillations are referred to as dispersive shocks. The Whitham equations for the mKdV equation are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the solutions of the Whitham equations when the initial values are given by a step function. We also compare the results with those of the defocusing nonlinear Schrodinger (NLS) equation. For the NLS equation, the Whitham equations are strictly hyperbolic and genuinely nonlinear. We show that the weak hyperbolicity of the mKdV-Whitham equations is responsible for an additional structure in the dispersive shocks which has not been found in the NLS case.