A new two-dimensional lattice model that is "consistent around a cube"

Abstract

For two-dimensional lattice equations one definition of integrability is that the model can be naturally and consistently extended to three dimensions, i.e., that it is "consistent around a cube" (CAC). As a consequence of CAC one can construct a Lax pair for the model. Recently Adler, Bobenko and Suris conducted a search based on this principle and certain additional assumptions. One of those assumptions was the "tetrahedron property", which is satisfied by most known equations. We present here one lattice equation that satisfies the consistency condition but does not have the tetrahedron property. Its Lax pair is also presented and some basic properties discussed.

8 pages in LaTeX

References in corpus (2)

• Classification of integrable equations on quad-graphs. The consistency approach
• Lax pair for the Adler (lattice Krichever-Novikov) System

Cited by in corpus (24)

• Continuous Symmetries of Difference Equations
• Classification of 3D consistent quad-equations
• Backlund transformations for integrable lattice equations
• Multiple-scale analysis of discrete nonlinear partial difference equations: the reduction of the lattice potential KdV
• Darboux Integrability of Trapezoidal $H^4$ and $H^6$ Families of Lattice Equations II: General Solutions
• Darboux integrability of trapezoidal $H^4$ and $H^6$ families of lattice equations I: First integrals
• Supersymmetric KdV equation: Darboux transformation and discrete systems
• Discrete Reductive Perturbation Technique
• Linear quadrilateral lattice equations and multidimensional consistency
• On Non-Point Invertible Transformations of Difference and Differential-Difference Equations
• Linearizability of Nonlinear Equations on a Quad-Graph by a Point, Two Points and Generalized Hopf-Cole Transformations
• Examples of Darboux Integrable Discrete Equations Possessing First Integrals of an Arbitrarily High Minimal Order
• On the Lagrangian structure of 3D consistent systems of asymmetric quad-equations
• Some incidence theorems and integrable discrete equations
• Extended Z-invariance for integrable vector and face models and multi-component integrable quad equations
• A Completeness Study on Certain $2\times 2$ Lax Pairs Including Zero Terms
• Discrete nonlinear hyperbolic equations. Classification of integrable cases
• Extension of the Adler-Bobenko-Suris classification of integrable lattice equations
• On multidimensional consistent systems of asymmetric quad-equations
• Reconstructing a Lattice Equation: a Non-Autonomous Approach to the Hietarinta Equation
• Linear degree growth in lattice equations
• On Darboux non-integrability of the Hietarinta equation
• On Initial Data in the Problem of Consistency on Cubic Lattices for $3\times 3$ Determinants
• Integrability conditions for nonautonomous quad-graph equations