Continuously symmetric solutions of the Adler-Bobenko-Suris class of discrete integrable equations are presented. Initially defined by their invariance under the action of both of the extended three point generalized symmetries admitted by the corresponding equations, these solutions are shown

We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler–Bobenko–Suris (ABS) list into the discrete Schrödinger spectral problem associated with Volterra-type equations. We show that the ABS equations correspond to Bäcklund

The classification of lattice equations that are integrable in the sense of higherdimensional consistency is extended by allowing directed edges. We find two cases that are not transformable via the ‘admissible transformations ’ to the lattice equations in the existing classification. PACS: 02.30.Ik, 04.60.Nc, 05.45.Yv 1

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-Bäcklund transformations and Lax pairs for all the equations in this class

Abstract. This paper deals with conservation laws for all integrable difference equations that belong to the famous Adler-Bobenko-Suris classification. All inequivalent three-point conservation laws are found, as are three five-point conservation laws for each equation. We also describe a method

The generalized symmetry method is applied to a class of completely discrete equations including the Adler-Bobenko-Suris list. Assuming the existence of a generalized symmetry, we derive a few integrability conditions suitable for testing and classifying equations of this class. Those conditions

This paper describes symmetries of all integrable difference equations that belong to the famous Adler-Bobenko-Suris classification. For each equation, the characteristics of symmetries satisfy a functional equation, which we solve by reducing it to a system of partial differential equations

We study the deformations of the H equations, presented recently by Adler, Bobenko and Suris, which are naturally defined on a black-white lattice. For each one of these equations, two different three-leg forms are constructed, leading to two different discrete Toda type equations

doi:10.3842/SIGMA.2010.033 Abstract. We use the classification of the quadrirational maps given by Adler, Bobenko and Suris to describe when such maps satisfy the Yang–Baxter relation. We show that the corresponding maps can be characterized by certain singularity invariance condition. This leads

there now exist classification results by Adler, Bobenko and Suris (ABS) leading to some new examples in addition to the lattice equations “of KdV type ” that were known since the late 1970s and early 1980s. In this paper we review the construction of soliton solutions for the KdV type lattice equations