Generalized mKdV Equation and Genus Two Jacobi Type Hyperelliptic Differential Equation

Abstract

We generalized the mKdV equation in order that the static equations include $\mathrm{sn}$ 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 $\hat{I}{»}_6=0,\hat{I}{»}_0=0$, Weierstrass type and Jacobi type hyperelliptic functions are different solutions to the same hyperelliptic differential equations. Then these solutions are connected by the special $\mathrm{Sp}(4,\mathbf{R})$ Lie group transformation.

10 pages