Affine and Finite Lie Algebras and Integrable Toda Field Equations on Discrete Space-Time

Difference-difference systems are suggested corresponding to the Cartan matrices of any simple or affine Lie algebra. In the cases of the algebras $A_N$, $B_N$, $C_N$, $G_2$, $D_3$, $A_1^{(1)}$, $A_2^{(2)}$, $D^{(2)}_N$ these systems are proved to be integrable. For the systems corresponding to the algebras $A_2$, $A_1^{(1)}$, $A_2^{(2)}$ generalized symmetries are found. For the systems $A_2$, $B_2$, $C_2$, $G_2$, $D_3$ complete sets of independent integrals are found. The Lax representation for the difference-difference systems corresponding to $A_N$, $B_N$, $C_N$, $A^{(1)}_1$, $D^{(2)}_N$ are presented

Classification of five-point differential-difference equations

Using the generalized symmetry method, we carry out, up to autonomous point transformations, the classification of integrable equations of a subclass of the autonomous five-point differential-difference equations. This subclass includes such well-known examples as the Itoh-Narita-Bogoyavlensky and the discrete Sawada-Kotera equations. The resulting list contains 17 equations some of which seem to be new. We have found non-point transformations relating most of the resulting equations among themselves and their generalized symmetries.Comment: 29 page

Integrable discrete autonomous quad-equations admitting, as generalized symmetries, known five-point differential-difference equations

In this paper we construct the autonomous quad-equations which admit as symmetries the five-point differential-difference equations belonging to known lists found by Garifullin, Yamilov and Levi. The obtained equations are classified up to autonomous point transformations and some simple non-autonomous transformations. We discuss our results in the framework of the known literature. There are among them a few new examples of both sine-Gordon and Liouville type equations.Comment: 27 page