Uniform Asymptotic Solutions of Discontinuous Initial Value Problems for Dispersive Hyperbolic Equations*

This paper is concerned with a discontinuous initial value problem for a dispersive hyperbolic equation where the dispersive term of the equation contains a large parameter. In the case of an equation with constant coefficients the initial value problem can be solved exactly leading to an integral representation which is expanded asymptotically. Nonuniform and uniform asymptotic expansions are obtained. The nonuniform expansion is singular on a certain space-time line. This singularity is similar to the anomaly occurring at a shadow-boundary in geometrical optics. For the equation with nonconstant coefficients nonuniform and uniform asymptotic solutions of the initial value problem are derived by ray methods. It is verified that these solutions are in perfect agreement with the nonuniform and uniform asymptotic expansions of the exact solution of the problem.