Integrable lattices and their sublattices II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices

TL;DR

This paper explores integrable self-adjoint schemes on triangular and honeycomb lattices, introduces the star-triangle relation, and constructs a new integrable discrete 3D system with geometric interpretation.

Contribution

It develops a sublattice approach to integrable schemes on complex lattices and introduces the star-triangle relation linking these systems, along with Darboux transformations.

Findings

Established integrable self-adjoint schemes on triangular and honeycomb lattices.

Introduced the star-triangle relation connecting these schemes.

Constructed a novel integrable discrete 3D system with geometric interpretation.

Abstract

An integrable self-adjoint 7-point scheme on the triangular lattice and an integrable self-adjoint scheme on the honeycomb lattice are studied using the sublattice approach. The star-triangle relation between these systems is introduced, and the Darboux transformations for both linear problems from the Moutard transformation of the B-(Moutard) quadrilateral lattice are obtained. A geometric interpretation of the Laplace transformations of the self-adjoint 7-point scheme is given and the corresponding novel integrable discrete 3D system is constructed.