Parametric Korteweg–de Vries Hierarchy and Hyperelliptic Sigma Functions

Abstract

In this paper, a parametric Korteweg–de Vries hierarchy is defined that depends on an infinite set of graded parameters \(a = (a_4,a_6,\dots)\). It is shown 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 \(t = (t_1, t_3,\dots, t_{2g-1})\) and \(\lambda = (\lambda_4, \lambda_6,\dots, \lambda_{4 g + 2})\), specifies a solution to this hierarchy in which the parameters \(a\) are given as polynomials in the parameters \(\lambda\) of the sigma function. The proof uses results concerning the family of operators introduced by V. M. Buchstaber and S. Yu. Shorina. This family consists of \(g\) third-order differential operators in \(g\) variables. Such families are defined for all \(g \geqslant 1\), the operators in each of them pairwise commute with each other and also commute with the Schrödinger operator. In this paper a relationship between these families and the Korteweg–de Vries parametric hierarchy is described. 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.