Toric networks, geometric $R$-matrices and generalized discrete Toda lattices

TL;DR

This paper introduces a new family of integrable systems generalizing the discrete Toda lattice using toric networks and geometric R-matrices, providing explicit solutions via algebraic geometry methods.

Contribution

It defines a three-parameter family of generalized discrete Toda lattices and constructs their integrals of motion, spectral map, and explicit solutions.

Findings

Constructed integrals of motion for the generalized system

Linearization of time evolutions on the Jacobian of the spectral curve

Explicit solutions using Riemann theta functions

Abstract

We use the combinatorics of toric networks and the double affine geometric $R$-matrix to define a three-parameter family of generalizations of the discrete Toda lattice. We construct the integrals of motion and a spectral map for this system. The family of commuting time evolutions arising from the action of the $R$-matrix is explicitly linearized on the Jacobian of the spectral curve. The solution to the initial value problem is constructed using Riemann theta functions.