Soliton Solutions of Korteweg-de Vries Equations and Hyperelliptic Sigma Functions

The modern soliton theories [DJKM, SN, SS], which were developed in ending of last century, are known as the infinite dimensional analysis and gave us fruitful and beautiful results, e.g., relations of soliton equations to universal Grassmannian manifold, Plücker embedding, infinite dimensional Lie algebra, loop algebra, representation theories, Schur functions, Young diagram, and so on. They stemmed from an investigation of the Bäcklund transformations among the soliton solutions, which are expressed by hyperbolic functions [DJKM, SN, SS]. By primitive considerations, the soliton solutions are related to certain degenerate algebraic curves and thus the Bäcklund transformation can be regarded as transformation among certain degenerate curves of different genera [DJKM, SN, SS]. As the modern soliton theories are based on the abstract theory, theories on the Abelian functions established in nineteenth century are very concrete [B1-3, Kl]. They are of given concrete algebraic curves and of Abelian functions and differential equations there. By fixing a hyperelliptic curve, Klein defined the hyperelliptic sigma function, a hyperelliptic version of Weierstrass sigma function for elliptic curve [Kl]. The sigma function is a well-tuned theta function and brings us fruitful information of the curve. In terms of the hyperelliptic sigma functions and bilinear differential operator, Baker discovered the Korteweg-de Vries hierarchy, Kadomtsev-Petviashvili equation and gave their periodic solutions without ambiguous parameters [B1-3, Ma1]. As the new century began, I believe that we should connect both established theories from a novel point of view. Recently the theories in nineteenth century has been re-evaluated in various fields from similar viewpoint [BEL1-5,CEEK, EE, EEL, EEP, Ma1,2, N]. For example, Buchstaber, Enolskii and Leykin connected the hyperelliptic sigma functions with the Paffian in [BEL3] and the Schur-Weierstrass polynomials in [BEL4]. The purpose of this article is also to give a step to a unification of both theories. In this article, we will focus on the soliton solutions of Korteweg-de Vries equation in terms of the hyperelliptic sigma functions over degenerate hyperelliptic curves. It is needless to say that the