The Davey-Stewartson equations : a numerical study

Abstract

In 1974 Davey and Stewartson used a multi-scale analysis to derive a coupled system of nonlinear partial differential equations which describes the evolution of a three dimensional wave packet in water of a finite depth. This system of equations is the closest integrable two dimensional analog of the well-known one dimensional nonlinear Schrodinger equation. The method of inverse scattering can thus be used to solve the Davey-Stewartson equations in theory, but in practice this method is not feasible with arbitrary initial conditions. In this thesis we present a numerical method for solving the Davey-Stewartson equations. It is an extension of the split-step Fourier method that proved to be so successful for the nonlinear Schrodinger equation. This method is tested on some known soliton and dromion solutions and then used to study modulational instability and solutions which become singular in finite time.