Hyperelliptic Solutions of KdV and KP equations: Reevaluation of Baker's Study on Hyperelliptic Sigma Functions

Abstract

Explicit function forms of hyperelliptic solutions of Korteweg-de Vries (KdV) and \break 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) {\bf{27}}, 135-156), H. F. Baker essentially derived KdV hierarchy and KP equation by using bilinear differential operator $\mathbf{D}$, identities of Pfaffians, symmetric functions, hyperelliptic $\sigma$-function and $\wp$-functions; ${\wp }_{\mu \nu }=-{\partial }_{\mu }{\partial }_{\nu }\log \sigma$ $=-({\mathbf{D}}_{\mu }{\mathbf{D}}_{\nu }\sigma \sigma )/2{\sigma }^2$. The connection between his theory and the modern soliton theory was also discussed.

AMS-Tex, 12 pages

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Cited by in corpus (16)

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• Hyperelliptic Loop Solitons with Genus g: Investigations of a Quantized Elastica
• Sigma, tau and Abelian functions of algebraic curves
• Hyperelliptic Solutions of Modified Korteweg-de Vries Equation of Genus g: Essentials of Miura Transformation
• Closed Loop Solitons and Sigma Functions: Classical and Quantized Elasticas with Genera One and Two
• Differential Equations of Genus Four Hyperelliptic $\wp$ Functions
• Elliptic and Hyperelliptic Solutions of Discrete Painlevé I and Its Extensions to Higher Order Difference Equations
• Construction of Two Parametric Deformation of KdV-Hierarchy and Solution in Terms of Meromorphic Functions on the Sigma Divisor of a Hyperelliptic Curve of Genus 3
• Toda Equations and $\sigma$-Functions of Genera One and Two
• Hyperelliptic sigma functions and the Kadomtsev-Petviashvili equation
• Genus Two Quasi-Siegel Modular Forms and Gromov-Witten Theory of Toric Calabi-Yau Threefolds
• The al function of a cyclic trigonal curve of genus three
• Sigma function associated with a hyperelliptic curve with two points at infinity
• Soliton Solutions of Kortweg-de Vries Equations and Hyperelliptic Sigma Functions
• Prime form and sigma function
• Quasi-periodic and periodic solutions of the Toda lattice via the hyperelliptic sigma function