Nonlinear evolutions equations in hirota's and sato's theories via young and maya diagrams

Abstract

This work relates Hirota direct method to Sato theory. The bilinear direct method was introduced by Hirota to obtain exact solutions for nonlinear evolution equations. This method is applied to the Kadomtsev-Petviashvili (KP), KortewegdeVries (KdV), Sawada-Kotera (S-K) and sine-Gordon (s-G) equations and solved to generate multi-soliton solutions. The Hirota’s scheme is shown to link to the Sato theory and later produced the Sato equation. It is also shown that the -function, which underlies the form of the soliton solutions, acts as the key function to express the solutions of the Sato equation. By using the results of group representation theory, particularly via Young and Maya diagrams, it is shown that the -function is naturally being governed by the class of physically significant nonlinear partial differential equations in the bilinear forms of Hirota scheme and are closely related to the Plucker relations. This framework is shown for Kadomtsev-Petviashvili (KP), KortewegdeVries (KdV), Sawada-Kotera (S-K) and sine-Gordon (s-G) equations.