Hyperelliptic solutions of KdV and KP equations: re-evaluation of Baker's study on hyperelliptic sigma functions

Abstract

Explicit function forms of hyperelliptic solutions of Korteweg-de Vries (KdV) and Kadomtsev-Petviashvili (KP) equations were constructed for a given curve y 2 = f(x) whose genus is three. This study was based upon the fact that about one hundred years ago (Acta Math. (1903) 27, 135-156), H. F. Baker essentially derived KdV hierarchy and KP equation by using bilinear differential operator D, identities of Paffians, symmetric functions, hyperelliptic σ-function and ℘-functions; ℘µν = −∂µ∂ν log σ = −(DµDνσσ)/2σ 2. The connection between his theory and the modern soliton was also discussed. In this article we will construct explicit function forms of hyperelliptic solutions of Korteweg-de Vries (KdV) and Kadomtsev-Petviashvili (KP) equations for a given curve y2 = f(x) whose genus is three, along the lines of the study of H. F. Baker’s sigma function [B1, B2, B3]. This construction means reevaluation of Baker’s studies on hyperelliptic functions which were done one hundred years