Integrability conditions for two-dimensional lattices

Abstract

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 $\frac{{\partial }^2}{\partial u_{n+1}\partial u_{n-1}}f(u_{n+1},u_n,u_{n-1})=C{e}^{\alpha u_n-\frac{\alpha m}{2}{u}_{n+1}-\frac{\alpha k}{2}{u}_{n-1}},$ where $C$ and $\alpha$ are constant parameters and $k,m$ are nonnegative integers.

References in corpus (5)

• On integrability of (2+1)-dimensional quasilinear systems
• Commutator identities on associative algebras and integrability of nonlinear pde's
• Characteristic Lie rings, finitely-generated modules and integrability conditions for 2+1 dimensional lattices
• On a classification algorithm of the integrable two-dimensional lattices via Lie-Rinehart algebras
• Darboux Integrability of Trapezoidal $H^4$ and $H^6$ Families of Lattice Equations II: General Solutions

Cited by in corpus (1)

• On a class of 2D integrable lattice equations