Hirota–Sato Formalism on Some Nonlinear Waves Equations

Abstract

This article demonstrates that Hirota’s direct method or scheme for solving nonlinear waves equation is linked to Sato theory, and eventually resulted in the Sato equation. This theoretical framework or simply the Hirota–Sato formalism also reveals that the τ – function, which underlies the analytic form of soliton solutions of theses physically significant nonlinear waves equations, shall acts as the key function to express the solutions of Sato equation. From representation theory of groups, it is shown that the τ – function in the bilinear forms of Hirota scheme are closely connected to the Plucker relations in Sato theory. Thus Hirota–Sato formalism provides a deeper understanding of soliton theory from a unified viewpoint. The Kadomtsev–Petviashvili (KP), Korteweg–de Vries (KdV) and Sawada–Kotera equations are used to verify this framework.