Perturbation Method for a Nonlinear Wave Modulation. I

In this paper we consider a system of nonlinear wave equations which admits, in a linear approximation, a planewave solution with high-frequency oscillation. Then, for the wave of small but finite amplitude, we investigate how slowly varying parts of the wave such as the amplitude are modulated by nonlinear self-interactions. A stretching transformation shows that, in the lowest order of an asymptotic expansion, the original system of equations can be reduced to a tractable, single, nonlinear equation to determine the amplitude modulation.