Generalized mKdV Equation and Genus Two Jacobi Type Hyperelliptic Differential Equation

TL;DR

This paper extends the mKdV equation to include static equations with sn differential equations, establishing links between hyperelliptic functions and Lie group transformations for genus two curves.

Contribution

It generalizes the mKdV equation to encompass hyperelliptic functions and explores their interrelations via Lie group transformations for genus two curves.

Findings

Derived differential equations for genus two hyperelliptic functions.

Established correspondence between Weierstrass and Jacobi hyperelliptic solutions.

Connected solutions through Sp(4, R) Lie group transformation.

Abstract

We generalized the mKdV equation in order that the static equations include ${\rm sn}$ differential equation. As a result, a good correspondence was obtained between the KdV equation and the mKdV equation.For general genus two hyperelliptic curves, we obtained differential equations for Weierstrass type and Jacobi type hyperelliptic functions. Considering the special case of $\lambda_6=0, \lambda_0=0$, Weierstrass type and Jacobi type hyperelliptic functions are different solutions to the same hyperelliptic differential equations. Then these solutions are connected by the special ${\rm Sp(4, {\bf R})}$ Lie group transformation.