Differential Equations of Genus Four Hyperelliptic $\wp$ Functions

TL;DR

This paper investigates the differential equations of hyperelliptic $ ext{wp}$ functions up to genus four, revealing the complexity and limitations of Hirota form in characterizing integrable models as genus increases.

Contribution

It extends the understanding of differential equations of hyperelliptic $ ext{wp}$ functions to higher genus, highlighting the emergence of non-Hirota form equations beyond genus two.

Findings

Differential equations for genus two can be written in Hirota form.

For genus more than two, additional KdV equations appear.

Beyond genus three, some equations cannot be expressed in Hirota form.

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.