The application of dressing method on nonlinear evolution equations

Abstract

It is well known that Korteweg-de Vries (KdV) equation can be solved by inverse scattering transform (IST). Numerous efforts have been made to extend the range of application of this method. The question now is how wide is the class of equations which are integrable by IST? It would be more useful if it could be extended or generalised to accommodate other equation. That's why we introduced dressing method which was proposed by Zakharov and Shabat (1974). The aim of dressing method is to generate integrable nonlinear equation and simultaneously its solution. We study this method on three integral operators and the differential operators. Besides that, we also study and list down all the properties which will be use together with operators in this method. In this dissertation, we choose only constant coefficient operator and scalar differential operator. We applied it to derive integrable nonlinear KdV and Kadomtsev-Petviashvili (KP) and thereafter we solve for exact solution.