Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency

Abstract

We propose the notion of integrable boundary in the context of discrete integrable systems on quad-graphs. The equation characterizing the boundary must satisfy a compatibility equation with the one characterizing the bulk that we called the three-dimensional (3D) boundary consistency. In comparison to the usual 3D consistency condition which is linked to a cube, our 3D boundary consistency condition lives on a half of a rhombic dodecahedron. The We provide a list of integrable boundaries associated to each quad-graph equation of the classification obtained by Adler, Bobenko and Suris. Then, the use of the term "integrable boundary" is justified by the facts that there are Bäcklund transformations and a zero curvature representation for systems with boundary satisfying our condition. We discuss the three-leg form of boundary equations, obtain associated discrete Toda-type models with boundary and recover previous results as particular cases. Finally, the connection between the 3D boundary consistency and the set-theoretical reflection equation is established.

References in corpus (4)

• Classification of 3D consistent quad-equations
• On Quadrirational Yang-Baxter Maps
• Set-theoretical reflection equation: Classification of reflection maps
• The lattice Schwarzian KdV equation and its symmetries

Cited by in corpus (6)

• On the $3D$ consistency of a Grassmann extended lattice Boussinesq system
• Entwining Yang-Baxter maps related to NLS type equations
• A discrete Darboux-Lax scheme for integrable difference equations
• Algebraic entropy computations for lattice equations: why initial value problems do matter
• Integrable boundary conditions for quad equations, open boundary reductions and integrable mappings
• Classification of integrable boundary equations for integrable quad-graph systems