Integrability and Symmetries of Difference Equations: the Adler-Bobenko-Suris Case

We consider the partial difference equations of the Adler-Bobenko-Suris classification, which are characterized as multidimensionally consistent. The latter property leads naturally to the construction of auto-B{\"a}cklund transformations and Lax pairs for all the equations in this class. Their symmetry analysis is presented and infinite hierarchies of generalized symmetries are explicitly constructed.Comment: 16 pages, for the proceedings of the 4th Workshop "Group Analysis of Differential Equations and Integrable Systems", Cyprus, 200

Darboux transformations with tetrahedral reduction group and related integrable systems

In this paper, we derive new two-component integrable differential difference and partial difference systems by applying a Lax-Darboux scheme to an operator formed from a

On consistent systems of difference equations

We consider overdetermined systems of difference equations for a single function u which are consistent, and propose a general framework for their analysis. The integrability of such systems is defined as the existence of higher order symmetries in both lattice directions and various examples are presented. Two hierarchies of consistent systems are constructed, the first one using lattice paths and the second one as a deformation of the former. These hierarchies are integrable and their symmetries are related via Miura transformations to the Bogoyavlensky and the discrete Sawada-Kotera lattices, respectively

Symmetry algebra of discrete KdV equations and corresponding differential-difference equations of Volterra type

A sequence of canonical conservation laws for all the Adler-Bobenko-Suris equations is derived and is employed in the construction of a hierarchy of master symmetries for equations H1-H3, Q1-Q3. For the discrete potential and Schwarzian KdV equations it is shown that their local generalized symmetries and non-local master symmetries in each lattice direction form centerless Virasoro type algebras. In particular, for the discrete potential KdV, the structure of its symmetry algebra is explicitly given. Interpreting the hierarchies of symmetries of equations H1-H3, Q1-Q3 as differential-difference equations of Yamilov's discretization of Krichever-Novikov equation, corresponding hierarchies of isospectral and non-isospectral zero curvature representations are derived for all of them.Comment: 22 page

Symmetries of ℤN graded discrete integrable systems

We recently introduced a class of ℤN graded discrete Lax pairs and studied the associated discrete integrable systems (lattice equations). We discuss differential–difference equations which then we interpret as symmetries of the discrete systems. In particular, we present nonlocal symmetries which are associated with the 2D Toda lattice

On the Lagrangian structure of 3D consistent systems of asymmetric quad-equations

Recently, the first-named author gave a classification of 3D consistent 6-tuples of quad-equations with the tetrahedron property; several novel asymmetric 6-tuples have been found. Due to 3D consistency, these 6-tuples can be extended to discrete integrable systems on Z^m. We establish Lagrangian structures and flip-invariance of the action functional for the class of discrete integrable systems involving equations for which some of the biquadratics are non-degenerate and some are degenerate. This class covers, among others, some of the above mentioned novel systems.Comment: 21 pp, pdfLaTe

Darboux transformation with dihedral reduction group

We construct the Darboux transformation with Dihedral reduction group for the 2-dimensional generalisation of the periodic Volterra lattice. The resulting Bäcklund transformation can be viewed as a nonevolutionary integrable differential difference equation. We also find its generalised symmetry and the Lax representation for this symmetry. Using formal diagonalisation of the Darboux matrix, we obtain local conservation laws of the system