论文标题
二维晶格的集成性条件
Integrability conditions for two-dimensional lattices
论文作者
论文摘要
在文章中，研究了$ u_ {n，xy} = f（u_ {n+1}，u_n，u_n，u_ {n-1}）$的非线性二维晶格的一些代数属性。详尽描述此类晶格的综合案例的问题仍然开放。通过使用该方法，在我们以前的作品中进行了开发和测试，我们对这种情况采用了特征性的lie-rinehart代数方法。在文章中，我们得出了晶格的有效可集成性条件，并证明在可集成的情况下，函数$ f（u_ {n+1}，u_n，u_ {n-1}）$是一个准polynomial满足以下等式$ \ frac $ \ frac {\ partial^2} {\ partial^2} {\ partial^2} { u_ {n-1}} f（u_ {n+1}，u_n，u_ {n-1}）= ce^{αu_n - {\ frac {\ frac {αm} {2} {2}} u_ {n+1}恒定参数和$ k，\，m $是非负整数。
In the article some algebraic properties of nonlinear two-dimensional lattices of the form $u_{n,xy} = f(u_{n+1}, u_n, u_{n-1})$ are studied. The problem of exhaustive description of the integrable cases of this kind lattices remains open. By using the approach, developed and tested in our previous works we adopted the method of characteristic Lie-Rinehart algebras to this case. In the article we derived an effective integrability conditions for the lattice and proved that in the integrable case the function $f(u_{n+1}, u_n, u_{n-1})$ is a quasi-polynomial satisfying the following equation $\frac{\partial^2}{\partial u_{n+1}\partial u_{n-1}}f(u_{n+1}, u_n, u_{n-1})=Ce^{αu_n-{\frac{αm}{2}}u_{n+1}-{\frac{αk}{2}}u_{n-1}},$ where $C$ and $α$ are constant parameters and $k,\,m$ are nonnegative integers.
