Classification of integrable boundary equations for integrable quad-graph systems

Abstract

In the context of integrable systems on quad-graphs, the boundary consistency around a half of a rhombic dodecahedron, as a companion notion to the three-dimensional consistency around a cube, was introduced as a criterion for defining integrable boundary conditions for quad-graph systems with a boundary. In this paper, we formalize the notions of boundary equations as boundary conditions for quad-graph systems, and provide a systematic method for solving the boundary consistency, which results in a classification of integrable boundary equations for quad-graph equations in the Adler-Bobenko-Suris classification. This relies on factorizing, first the quad-graph equations into pairs of dual boundary equations, and then the consistency on a rhombic dodecahedron into two equivalent boundary consistencies. Generalizations of the method to rhombic-symmetric equations are also considered.

arXiv commentv2, minor corrections, 35 pages, 13 Figures, 8 Tables

References in corpus (16)

• Bäcklund transformation for the Krichever-Novikov equation
• Yang-Baxter maps and symmetries of integrable equations on quad-graphs
• Soliton Solutions for ABS Lattice Equations II: Casoratians and Bilinearization
• Classification of 3D consistent quad-equations
• Boussinesq-like multi-component lattice equations and multi-dimensional consistency
• On Quadrirational Yang-Baxter Maps
• Integrability and Symmetries of Difference Equations: the Adler-Bobenko-Suris Case
• Set-theoretical reflection equation: Classification of reflection maps
• Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency
• Singularities of Type-Q ABS Equations
• Discrete Riemann surfaces based on quadrilateral cellular decompositions
• The asymptotic behaviour of the discrete holomorphic map $Z^a$ via the Riemann-Hilbert method
• Classification of quad-equations on a cuboctahedron
• On an inverse scattering transform for the nonlinear Schödinger equation on the half-line
• Integrable boundary conditions for quad equations, open boundary reductions and integrable mappings
• New Class of Integrable Maps of the Plane: Manin Transformations with Involution Curves