On the Whitham Equations for the Defocusing Complex Modified KdV Equation

Authors:
  • YKYuji Kodama
    Yuji Kodama
  • VPV. U. Pierce
    V. U. Pierce
  • FTFei-Ran Tian
    Fei-Ran Tian
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Abstract
We study the Whitham equations for the defocusing complex modified KdV (mKdV) equation. These Whitham equations are quasi-linear hyperbolic equations and 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-like function. We also compare the results with those of the defocusing nonlinear Schrödinger (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 some new structure in the dispersive shocks which has not been found in the NLS case.

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