On the Modulations and Nonlinear Interactions of Waves and the Nonlinear Stability of a Near‐Critical System

Abstract

The nonlinear interactions and modulations of an n‐dimensional wave and of a disturbance to a near‐critical system governed by a general (n + 1)‐dimensional system of equations are studied by perturbation methods. It is found that these modulations are governed by an evolution equation which is either by itself or coupled to a second equation, depending on the nature of the long wave solutions of the corresponding linearized system. When a single evolution equation exists, its leading terms are shown to give the nonlinear Schrödinger equation. Water waves and near‐critical plane Poiseuille flow are discussed as examples.