APPROXIMATION OF HIGH-FREQUENCY WAVE PROPAGATION IN DISPERSIVE MEDIA

We consider semilinear hyperbolic systems with a trilinear nonlinearity. Both the differential equation and the initial data contain the inverse of a small parameter E, and typical solutions oscillate with frequency proportional to 1/E in time and space. Moreover, solutions have to be computed on time intervals with length of 1/E in order to study nonlinear and diffractive effects. As a consequence, direct numerical simulations are extremely costly or even impossible. We propose an analytical approximation and prove that it approximates the exact solution up to an error of O(E2) over times of 1/E. This is a significant improvement over the classical nonlinear Schro"\dinger approximation, which only yields an accuracy of O(E).