B\"acklund Transformation for the BC-Type Toda Lattice

TL;DR

This paper develops a Bäcklund transformation for a boundary-parameterized open Toda lattice, establishing its key properties and linking it to discretized dynamics using dual Lax matrices sharing the same spectral curve.

Contribution

It introduces a novel Bäcklund transformation for the BC-type Toda lattice with boundary parameters, demonstrating its canonicity, commutativity, and spectrality, and explores its interpretation as discretized time evolution.

Findings

Constructed Bäcklund transformation with boundary parameters

Proved canonicity, commutativity, and spectrality of the transformation

Linked the transformation to discretized time dynamics using dual Lax matrices

Abstract

We study an integrable case of n-particle Toda lattice: open chain with boundary terms containing 4 parameters. For this model we construct a B\"acklund transformation and prove its basic properties: canonicity, commutativity and spectrality. The B\"acklund transformation can be also viewed as a discretized time dynamics. Two Lax matrices are used: of order 2 and of order 2n+2, which are mutually dual, sharing the same spectral curve.