On the Whitham Equations for the Semiclassical Limit of the Defocusing Nonlinear Schrodinger Equation

摘要

We study the Whitham equations, which describe the semiclassical limit of the defocusing nonlinear Schrodinger 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.