Direct linearisation of the discrete-time two-dimensional Toda lattices

TL;DR

This paper develops a direct linearisation approach for the discrete-time two-dimensional Toda lattice of $A_ infty$-type, enabling the analysis of multiple nonlinear equations and the construction of general solutions, including periodic reductions.

Contribution

It introduces a direct linearisation framework for the discrete-time 2D Toda lattice, allowing simultaneous treatment of several nonlinear equations and their solutions, and explores their periodic reductions.

Findings

Established a linearisation method for the $A_ infty$-type lattice.

Derived solutions for the Toda lattice and its reductions.

Analyzed integrability properties of the reduced models.

Abstract

The discrete-time two-dimensional Toda lattice of $A_\infty$-type is studied within the direct linearisation framework, which allows us to deal with several nonlinear equations in this class simultaneously and to construct more general solutions of these equations. The periodic reductions of this model are also considered, giving rise to the discrete-time two-dimensional Toda lattices of $A_{r-1}^{(1)}$-type for $r\geq 2$ (which amount to the negative flows of members in the discrete Gel'fand--Dikii hierarchy) and their integrability properties.