A Perturbation Method for Hyperbolic Equations with Small Nonlinearities

Abstract

A method of multiple scales is developed for the generation of uniformly valid asymptotic solutions of initial value problems for nonlinear wave equations. The method is applicable when the nonlinearities are small and only involve the first derivatives of the dependent variable. The determination of the first approximation is reduced to the solution of a pair of nonlinear, first order, ordinary differential equations. The method is then extended to apply to canonical systems of hyperbolic equations, and to problems in which the nonlinearities also involve the second derivatives of the dependent variable and hence lead to shocks.