Parametric Korteweg--de Vries hierarchy and hyperelliptic sigma functions

TL;DR

This paper introduces a parametric Korteweg--de Vries hierarchy linked to hyperelliptic sigma functions, showing solutions can be expressed via Kleinian functions and exploring related differential operator families.

Contribution

It defines a new parametric hierarchy dependent on infinite parameters and relates it to hyperelliptic sigma functions and differential operator families, extending previous operator constructions.

Findings

Solutions of the hierarchy are given by Kleinian functions derived from sigma functions.

The paper establishes a relationship between the hierarchy and commuting differential operators.

Extensions to infinite families of differential operators are constructed and analyzed.

Abstract

In this paper we define the parametric Korteweg-de Vries hierarchy that depends on an infinite set of graded parameters $a = (a_4,a_6,\dots)$. We show that, for any genus $g$, the Klein hyperelliptic function $\wp_{1,1}(t,\lambda)$ defined on the basis of the multidimensional sigma function $\sigma(t, \lambda)$, where $t = (t_1, t_3,\dots, t_{2g-1})$, $\lambda = (\lambda_4, \lambda_6,\dots, \lambda_{4 g + 2})$, determines a solution of this hierarchy, where the parameters $a$ are given as polynomials in the parameters $\lambda$ of the sigma function. The proof uses results on the family of operators introduced by V. M. Buchstaber and S. Yu. Shorina. This family consists of $g$ third-order differential operators of $g$ variables. Such families are defined for all $g \geqslant 1$, the operators in each of them commute in pairs and also commute with the Schr\"odinger operator. In this paper, we describe the relationship between these families and the parametric Korteweg--de Vries hierarchy. A similar infinite family of third-order operators on an infinite set of variables is constructed. The results obtained are extended to the case of such a family.