Title: KdV hierarchies and quantum Novikov's equations Show Chinese title

Title: KdV层次结构和量子诺维科夫方程

Abstract: The paper begins with a review of the well known Novikov's equations and corresponding finite KdV hierarchies. For a positive integer $N$ we give an explicit description of the $N$-th Novikov's equation and its first integrals. Its finite KdV hierarchy consists of $N$ compatible integrable polynomial dynamical systems in $\mathbb{C}^{2N}$. Then we discuss a non-commutative version of the $N$-th Novikov's equation defined on a finitely generated free associative algebra $\mathfrak{B}_N$ with $2N$ generators. In $\mathfrak{B}_N$, for $N=1,2,3,4$, we have found two-sided homogeneous ideals $\mathfrak{Q}_N\subset\mathfrak{B}_N$ (quantisation ideals) which are invariant with respect to the $N$-th Novikov's equation and such that the quotient algebra $\mathfrak{C}_N = \mathfrak{B}_N\diagup \mathfrak{Q}_N$ has a well defined Poincare-Birkhoff-Witt basis. It enables us to define the quantum $N$-th Novikov's equation on the $\mathfrak{C}_N$. We have shown that the quantum $N$-th Novikov's equation and its finite hierarchy can be written in the standard Heisenberg form. Show Chinese abstract

Abstract: 论文首先回顾了著名的诺维科夫方程及其相应的有限KdV层次结构。 对于正整数$N$，我们给出了第$N$个诺维科夫方程及其首次积分的显式描述。 其有限KdV层次结构包含$N$个在$\mathbb{C}^{2N}$上相容的可积多项式动力系统。 然后我们讨论在有限生成自由结合代数$\mathfrak{B}_N$上定义的第$N$个诺维科夫方程的非交换版本，该代数有$2N$个生成元。 在$\mathfrak{B}_N$中，对于$N=1,2,3,4$，我们找到了两个-sided homogeneous ideals$\mathfrak{Q}_N\subset\mathfrak{B}_N$（quantisation ideals），它们相对于$N$-th Novikov's equation 是不变的，并且商代数$\mathfrak{C}_N = \mathfrak{B}_N\diagup \mathfrak{Q}_N$具有明确的 Poincare-Birkhoff-Witt 基。 这使我们能够在$\mathfrak{C}_N$上定义量子$N$-th Novikov's equation。 我们已经证明，量子$N$-th 诺维科夫方程及其有限层次可以写成标准的海森堡形式。