Generalized mKdV Equation and Genus Two Jacobi Type Hyperelliptic Differential Equation

TL;DR

The paper addresses extending the mKdV equation to include the differential equation for ${\rm sn}$ to establish a close correspondence with the KdV equation and to develop a genus-two hyperelliptic function framework. It demonstrates integrability of the generalized mKdV via the AKNS/Wadati formalism and derives multiple static-transformation relations (Miura, square, inverse square, inverse power) that connect generalized mKdV and KdV. It then develops differential equations for genus-two Weierstrass-type functions $\wp_{22},\wp_{21},Q$ and their Jacobi-type duals $\widehat{\wp}_{11},\widehat{\wp}_{21},\widehat{Q}$, including a generalized Kummer surface relation and a half-period ${\rm Sp}(4,\mathbb{R})$ transformation linking the two in special cases. The results broaden the integrable structure for nonlinear PDEs and provide a coherent framework for analyzing multi-periodic genus-two hyperelliptic solutions and their interrelations.

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 $λ_6=0, λ_0=0$, 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.