Integrable dynamics of Toda-type on the square and triangular lattices

TL;DR

This paper develops new integrable Toda-type dynamics on square and triangular lattices, expanding the class of solvable lattice models with associated tau-functions and Bäcklund transformations.

Contribution

It introduces additional integrable Toda-type models on different lattices, along with their tau-function formulations and Darboux-Bäcklund transformations.

Findings

New integrable dynamics on square and triangular lattices

Explicit tau-function and Bäcklund transformation constructions

Enhanced understanding of nonlinear symmetries in lattice equations

Abstract

In a recent paper we constructed an integrable generalization of the Toda law on the square lattice. In this paper we construct other examples of integrable dynamics of Toda-type on the square lattice, as well as on the triangular lattice, as nonlinear symmetries of the discrete Laplace equations on the square and triangular lattices. We also construct the $\tau$ - function formulations and the Darboux-B\"acklund transformations of these novel dynamics.