We study the multiphases in the KdV zero-dispersion limit. These phases are governed by the Whitham equations, which are 2g + 1 quasi-linear hyperbolic equations where g is the number of phases. We are interested in both the interaction of two single phases and the breaking of a single phase for general initial data. We analyze in detail how a double phase is generated from the interaction or breaking, how it propagates in space-time, and how it collapses to a single phase in a finite time.The Whitham equations are known to be integrable via a hodograph transform. The crucial step in our approach is to formulate the hodograph transform in terms of the Euler-Poisson-Darboux solutions. Under our scheme, the zeros of the Jacobian of the transform are given by the zeros of the Euler-PoissonDarboux solution. Hence, the problem of inverting the hodograph transform to give the Whitham solution reduces to that of counting the zeros of the EulerPoisson-Darboux solution.

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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 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 an additional structure in the dispersive shocks which has not been found in the NLS case.

We study the Whitham equations, which describe the semiclassical limit of the defocusing nonlinear Schrödinger equation. The limit is governed by a pair of hyperbolic equations of two independent variables for a short time starting from the initial time. After this hyperbolic solution breaks down, the limit is described by the Whitham equations, which are four hyperbolic equations of two independent variables. We are interested in the evolution of the solutions from the pair of hyperbolic equations to Whitham equations. We use hodograph methods to solve the pair of hyperbolic equations and Whitham equations. Under our scheme, both are transformed to linear equations of Euler‐Poisson‐Darboux type whose solutions can be written down explicitly. We are able to establish the short‐time existence of the Whitham solution for generic initial data. Global (in time) results are also obtained for rather special initial data. © 1999 John Wiley & Sons, Inc.

Abstract. We study the Whitham equations for the fifth order KdV equation. The equations are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the solution of the Whitham equations when the initial values are given by a step function. We classify the step-like initial data into eight different types. We construct self-similar solutions for each type.