Differential Equations of Genus Four Hyperelliptic $\wp$ Functions

Abstract

In order to find higher dimensional integrable models, we study differential equations of hyperelliptic $\wp$ functions up to genus four. For genus two, differential equations of hyperelliptic $\wp$ functions can be written in the Hirota form. If the genus is more than one, we have KdV equation. If the genus is more than two, we have KdV and another KdV equations. If the genus becomes more than three, there appear differential equations which cannot be written in the Hirota form, which means that the Hirota form is not enough to characterize the integrable differential equations. We have shown that some differential equations are satisfied for general genus. We can obtain differential equations for general genus step by step.

arXiv comment19 pages

References in corpus (6)

• On a Ring of Formal Pseudo-differential Operators
• Hyperelliptic Solutions of KdV and KP equations: Reevaluation of Baker's Study on Hyperelliptic Sigma Functions
• The Static Elliptic $N$-soliton Solutions of the KdV Equation
• The Construction of the mKdV Cyclic Symmetric $N$-soliton Solution by the Bäcklund Transformation
• Common Hirota Form Bäcklund Transformation for the Unified Soliton System
• The Elliptic Function in Statistical Integrable Models II

Cited by in corpus (3)

• Hyperelliptic sigma functions and the Kadomtsev-Petviashvili equation
• Two Flows Kowalevski Top as the Full Genus Two Jacobi's Inversion Problem and Sp(4,$\mathbb{R}$) Lie Group Structure
• The Half-period Addition Formulae for Genus Two Hyperelliptic $\wp$ Functions and the Sp(4,$\mathbb{R}$) Lie Group Structure