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

Explicit function forms of hyperelliptic solutions of Korteweg-de Vries (Kd V) and Kadomtsev-Petviashvili (KP) equations are constructed for a given cu rve y(2) = f(x) whose genus is three. This paper is based upon the fact tha t about one hundred years ago (Baker H F 1903 Acta Math. 27 135-56;), Baker essentially derived KdV hierarchy and KP equations by using a bilinear dif ferential operator D, identities of Pfaffians,symmetric functions, the hype relliptic a-function and p-functions; p(uv) = -partial derivative (mu)parti al derivative (nu) log sigma = -(D(mu)D(nu)sigma sigma)/2 sigma (2). The co nnection between his theory and the modern soliton theory is also discussed .