Hirota bilinear method for nonlinear evolution equations

Abstract

The bilinear method introduced by Hirota to obtain exact solutions for nonlinear evolution equations is discussed. Firstly, several examples including the Korteweg-deVries, nonlinear Schrödinger and Toda equations are given to show how solutions are derived. Then after considering multi-dimensional systems such as the Kadomtsev-Petviashvili, two dimensional Toda and Hirota-Miwa equations, the algebraic structure of such nonlinear evolution systems is explained. Finally, extensions of the method including q-analogue, ultra-discrete systems and trilinear forms are also presented.