Infinitely many symmetries and conservation laws for quad-graph equations via the Gardner method

The application of the Gardner method for generation of conservation laws to all the ABS equations is considered. It is shown that all the necessary information for the application of the Gardner method, namely B\"acklund transformations and initial conservation laws, follow from the multidimensional consistency of ABS equations. We also apply the Gardner method to an asymmetric equation which is not included in the ABS classification. An analog of the Gardner method for generation of symmetries is developed and applied to discrete KdV. It can also be applied to all the other ABS equations

Yang-Baxter maps and multi-field integrable lattice equations

A variety of Yang-Baxter maps are obtained from integrable multi-field equations on quad-graphs. A systematic framework for investigating this connection relies on the symmetry groups of the equations. The method is applied to lattice equations introduced by Adler and Yamilov and which are related to the nonlinear superposition formulae for the B\"acklund transformations of the nonlinear Schr\"odinger system and specific ferromagnetic models.Comment: 16 pages, 4 figures, corrected versio

Yang Baxter maps with first degree polynomial 2 by 2 Lax matrices

A family of nonparametric Yang Baxter (YB) maps is constructed by refactorization of the product of two 2 by 2 matrix polynomials of first degree. These maps are Poisson with respect to the Sklyanin bracket. For each Casimir function a parametric Poisson YB map is generated by reduction on the corresponding level set. By considering a complete set of Casimir functions symplectic multiparametric YB maps are derived. These maps are quadrirational with explicit formulae in terms of matrix operations. Their Lax matrices are, by construction, 2 by 2 first degree polynomial in the spectral parameter and are classified by Jordan normal form of the leading term. Nonquadrirational parametric YB maps constructed as limits of the quadrirational ones are connected to known integrable systems on quad graphs

An algebraic method of classification of S-integrable discrete models

A method of classification of integrable equations on quad-graphs is discussed based on algebraic ideas. We assign a Lie ring to the equation and study the function describing the dimensions of linear spaces spanned by multiple commutators of the ring generators. For the generic case this function grows exponentially. Examples show that for integrable equations it grows slower. We propose a classification scheme based on this observation.Comment: 11 pages, workshop "Nonlinear Physics. Theory and Experiment VI", submitted to TM

Affine linear and D4 symmetric lattice equations: symmetry analysis and reductions

We consider lattice equations on Z2 which are autonomous, affine linear and possess the symmetries of the square. Some basic properties of equations of this type are derived, as well as a sufficient linearization condition and a conservation law. A systematic analysis of the Lie point and the generalized three- and five-point symmetries is presented. It leads to the generic form of the symmetry generators of all the equations in this class, which satisfy a certain non-degeneracy condition. Finally, symmetry reductions of certain lattice equations to discrete analogs of the Painlevé equations are considered