arXiv 1808.045622018-08-14nlin.SI

可积拟线性二维格的代数性质

Algebraic Properties of Quasilinear Two-Dimensional Lattices connected with integrability

I. T. Habibullin, M. N Kuznetsova

备注：: Submitted to Ufa Mathematical Journal

摘要：

基于可积约化的概念，讨论了三个自变量的非线性可积方程的一种分类方法。我们称该方程可积，如果它容许一大类约化为两个独立变量的双曲型方程的达布可积系统。最自然和方便的对象，在这个框架内的研究计划是类的二维格推广著名的户田格。本文讨论了$u_{n,xy}=α(u_{n+1} ,u_n,u_{n-1} )u_{n,x}u_{n,y} + β(u_{n+1},u_n,u_{n-1})u_{n,x}+γ(u_{n+1} ,u_n,u_{n-1} )u_{n,y}+δ(u_{n+1} ,u_n,u_{n-1})$形式的拟线性格。我们指定的系数的晶格假设存在切断条件，减少晶格的达布可积双曲型系统的任意高阶。在非简并性的一些额外假设下，我们描述了在上述意义下可积的格类。在获得的链列表中有新的例子。

英文摘要：

In the article a classification method for nonlinear integrable equations with three independent variables is discussed based on the notion of the integrable reductions. We call the equation integrable if it admits a large class of reductions being Darboux integrable systems of hyperbolic type equations with two independent variables. The most natural and convenient object to be studied within the frame of this scheme is the class of two dimensional lattices generalizing the well-known Toda lattice. In the present article we deal with the quasilinear lattices of the form $u_{n,xy}=α(u_{n+1} ,u_n,u_{n-1} )u_{n,x}u_{n,y} + β(u_{n+1},u_n,u_{n-1})u_{n,x}+γ(u_{n+1} ,u_n,u_{n-1} )u_{n,y}+δ(u_{n+1} ,u_n,u_{n-1})$. We specify the coefficients of the lattice assuming that there exist cutting off conditions which reduce the lattice to a Darboux integrable hyperbolic type system of the arbitrarily high order. Under some extra assumption of nondegeneracy we described the class of the lattices integrable in the sense indicated above. There are new examples in the obtained list of chains.

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英文摘要：

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