Reductive Perturbation Method and Far Fields of Wave Equations

Abstract

The concept of the far field is explained by means of the method of characteristics for the hyperbolic system of nonlinear partial differential equations. The reductive perturbation method is then presented as a method of solution which enables us to reduce a general nonlinear hyperbolic system to a single solvable nonlinear equation describing a far field of the system. The reductive perturbation method applied to more general systems including dissipation or dispersion shows that for long waves they can be reduced to the Burgers equation or the Kortweg-de Vries equation, respectively. Also, it is shown that a far field for the propagation of slow modulation of a plane wave of infinitesimal amplitude is governed by the Schrödinger equation and that for small but finite amplitudes a general wave system can be reduced to a nonlinear equation of Schrödinger type which we call the generalized nonlinear Schrödinger equation.