2D Toda Chain, Commuting Difference Operators and Holomorphic Bundles

TL;DR

This paper constructs high rank solutions to the 2D Toda lattice system by analyzing the discrete dynamics of Tyurin parameters, linking integrable systems with holomorphic vector bundles over algebraic curves.

Contribution

It introduces a novel method for constructing solutions to the 2D Toda system using the dynamics of Tyurin parameters and computes coefficients of commuting difference operators.

Findings

High rank solutions to the 2D Toda lattice are explicitly constructed.

Coefficients of high rank commuting difference operators are effectively calculated.

The approach connects integrable systems with the geometry of holomorphic vector bundles.

Abstract

High rank solutions to the 2D Toda Lattice System are constructed simultaneously with the effective calculation of coefficients of the high rank commuting ordinary difference operators. Our technic is based on the study of discrete dynamics of Tyurin Parameters characterizing the stable holomorphic vector bundles over the algebraic curves (Riemann Surfaces).