Characteristic Algebras of Fully Discrete Hyperbolic Type Equations

The notion of the characteristic Lie algebra of the discrete hyperbolic type equation is introduced. An effective algorithm to compute the algebra for the equation given is suggested. Examples and further applications are discussed.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Discretization of Liouville type nonautonomous equations preserving integrals

The problem of constructing semi-discrete integrable analogues of the Liouville type integrable PDE is discussed. We call the semi-discrete equation a discretization of the Liouville type PDE if these two equations have a common integral. For the Liouville type integrable equations from the well-known Goursat list for which the integrals of minimal order are of the order less than or equal to two we presented a list of corresponding semi-discrete versions. The list contains new examples of non-autonomous Darboux integrable chains.Comment: 27 page

Characteristic Lie Algebra and Classification of Semi-Discrete Models

Characteristic Lie algebras of semi-discrete chains are studied. The attempt to adopt this notion to the classification of Darboux integrable chains has been undertaken.Comment: 33 pages, corrected typos, submitted to the Proceedings of the workshop "Nonlinear Physics: Theory and Experiment IV", Theoretical Mathematical Physic

Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings

The main goal of the article is testing a new classification algorithm. To this end we apply it to a relevant problem of describing the integrable cases of a subclass of two-dimensional lattices. By imposing the cut-off conditions $u_{-1}=c_0$ and $u_{N+1}=c_1$ we reduce the lattice $u_{n,xy}=\alpha(u_{n+1},u_n,u_{n-1})u_{n,x}u_{n,y}$ to a finite system of hyperbolic type PDE. Assuming that for each natural $N$ the obtained system is integrable in the sense of Darboux we look for $\alpha$. To detect the Darboux integrability of the hyperbolic type system we use an algebraic criterion of Darboux integrability which claims that the characteristic Lie rings of such a system must be of finite dimension. We prove that up to the point transformations only one lattice in the studied class passes the test. The lattice coincides with the earlier found Ferapontov-Shabat-Yamilov equation. The one-dimensional reduction $x=y$ of this lattice passes also the symmetry integrability test

On Some Algebraic Properties of Semi-Discrete Hyperbolic Type Equations

Nonlinear semi-discrete equations of the form t_x(n+1)=f(t(n), t(n+1), t_x(n)) are studied. An adequate algebraic formulation of the Darboux integrability is discussed and the attempt to adopt this notion to the classification of Darboux integrable chains has been undertaken.Comment: 18 page

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