Geometric, algebraic and analytic properties of hyperelliptic $\mathrm{al}_{ab}$ function of genus $g$

TL;DR

This paper explores the properties of hyperelliptic al functions, generalizing elliptic functions, and demonstrates their role in formulating integrable nonlinear differential equations like the nonlinear Schrödinger and modified KdV equations.

Contribution

It introduces new differential identities of hyperelliptic al functions, extending solutions of integrable equations to the hyperelliptic case.

Findings

Derived differential identities for hyperelliptic al functions.

Extended hyperelliptic solutions to nonlinear Schrödinger and mKdV equations.

Identified the algebraic and geometric properties of these functions.

Abstract

In this paper, we investigate the geometric, algebraic and analytic properties of the hyperelliptic $\mathrm{al}_{ab}$ functions of a hyperelliptic curve $X$ with genus $g$ as the $\mathrm{al}_{ab}$ functions together with the $\mathrm{al}_a$ functions are a generalization of the Jacobi elliptic $\mathrm{sn}$, $\mathrm{cn}$, and $\mathrm{dn}$ functions. We then demonstrate the differential identities of the $\mathrm{al}_{ab}$ function. These identities are the novel integrable partial nonlinear differential equations as a natural extension of the hyperelliptic solutions of the modified Korteweg-de Vries equation in terms of the $\mathrm{al}_a$ function. Thus, we also show that by the identities, the $\mathrm{al}_{ab}$ function has the capability to be the hyperelliptic solution to the nonlinear Schr\"odinger and complex modified Korteweg-de Vries equations.