Classification of Integrable Boundary Equations for Integrable Systems on Quad-Graphs

Abstract

AbstractIn 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. Generalization of the method to rhombic-symmetric equations is also considered.