论文标题
参数korteweg- de vries层次结构和透明sigma函数
Parametric Korteweg--de Vries hierarchy and hyperelliptic sigma functions
论文作者
论文摘要
在本文中，我们定义了参数korteweg-de vries层次结构，该层次结构取决于无限的分级参数$ a =（a_4，a_6，\ dots）$。我们表明，对于任何属$ g $，klein klein hypelliptic函数$ \ wp_ {1,1}（t，λ）$根据多维Sigma函数$σ（t，λ）$，$ t =（$ t =（t_1，t_1，t_3，t_3，t_3，t_ dots，t_ d _ {2g-1） λ_{4 g + 2}）$，确定该层次结构的解决方案，其中参数$ a $以sigma函数的参数为$λ$作为多项式给出。 该证明使用了V. M. Buchstaber和S. Yu引入的运营商家族的结果。肖琳娜。该家庭由$ g $变量的$ g $三阶差速器运营商组成。为所有$ g \ geqslant 1 $定义了这样的家庭，每个运营商都成对上下班，并与Schrödinger运营商通勤。 在本文中，我们描述了这些家庭与参数Korteweg--de Vries层次结构之间的关系。构建了一组无限变量的类似的无限三阶运算符家族。获得的结果扩展到了这样一个家庭的情况。
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.
