Elliptic analog of the Toda lattice

TL;DR

This paper constructs action-angle variables for an elliptic Toda lattice system, solving it explicitly using Riemann theta functions and linking it to pole dynamics of elliptic solutions of the 2D Toda lattice.

Contribution

It introduces an elliptic analog of the Toda lattice, providing explicit solutions and connecting the system to spectral curves and elliptic functions.

Findings

Constructed action-angle variables for the elliptic Toda system.

Solved the system explicitly using Riemann theta functions.

Linked the system to pole dynamics of elliptic solutions of 2D Toda lattice.

Abstract

The action-angle variables for N-particle Hamiltonian system with the Hamiltonian $H=\sum_{n=0}^{N-1} \ln sh^{-2}(p_n/2)+\ln(\wp(x_n-x_{n+1})- \wp(x_n+x_{n+1})), x_N=x_0,$ are constructed, and the system is solved in terms of the Riemann $\theta$-functions. It is shown that this system describes pole dynamics of the elliptic solutions of 2D Toda lattice corresponding to spectral curves defined by the equation $w^2-P_{N}^{el}(z)w+\Lambda^{2N}=0$, where $P_{N}^{el}(z)$ is an elliptic function with pole of order N at the point z=0.