arXiv 2209.122862022-09-25nlin.SI

参数Korteweg-de Vries族与超椭圆sigma函数

Parametric Korteweg--de Vries hierarchy and hyperelliptic sigma functions

E. Yu. Bunkova, V. M. Buchstaber

DOI：: 10.4213/faa4020

摘要：

在本文中，我们定义的参数Korteweg-de Vries层次依赖于一个无限集的分级参数$a = (a_4,a_6,\dots)$。我们表明，对于任何属$g$，克莱因超椭圆函数$\wp_{1,1}(t,λ)$定义的多维西格玛函数$σ(t, λ)$的基础上，其中$t = (t_1, t_3,\dots, t_{2g-1})$，$λ= (λ_4, λ_6,\dots, λ_{4 g + 2})$，确定了解决方案的层次结构，其中参数$a$给出的多项式中的参数$λ$的西格玛函数。证明使用了V.M. Buchstaber和S. Yu. Shorina这个族由$g$变量的$g$三阶微分算子组成。这样的家庭被定义为所有的$g \geqslant 1$，在他们每个人的运营商交换成对，也与薛定谔运营商交换。在本文中，我们描述了这些家庭和参数Korteweg-德弗里斯层次之间的关系。构造了无穷多个变量上的一个类似的三阶算子无穷族。所得到的结果被推广到这样一个家庭的情况下。

英文摘要：

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 $\wp_{1,1}(t,λ)$ defined on the basis of the multidimensional sigma function $σ(t, λ)$, where $t = (t_1, t_3,\dots, t_{2g-1})$, $λ= (λ_4, λ_6,\dots, λ_{4 g + 2})$, determines a solution of this hierarchy, where the parameters $a$ are given as polynomials in the parameters $λ$ 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 Schrödinger 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.

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