On Nonlinear Evolution Equations with Applications
This paper is concerned with the most general water wave equations in (3 + 1) dimensions. Special attention is given to the derivation of the (1 + 1)-dimensional forced Korteweg-de Vries (fKdV) equation and the (1 + 1)-dimensional forced nonlinear Schrödinger (fNLS) equation near resonant conditions. Using the multiple scale technique, the (1 + 1)-dimensional nonlinear Schrödinger (NLS) equation for the amplitude function A0 of wake packets is derived. Section 6 deals with the fourth-order nonlinear Schrödinger equation for the amplitude of the wave potential leading to a major improvement in agreement with Longuet-Higgins analysis (1978a, b). A stability analysis is made based on the simplified version of the Dysthe's (1979) NLS equation. This is followed by Hogan's analysis of the fourth-order evolution equation for capillary-gravity waves in deep water. The final section deals with the Davey-Stewartson equations and the Kadomtsev and Petviashvili equation in water of finite depth.
