Elliptic analog of the Toda lattice

Abstract

The action-angle variables for N-particle Hamiltonian system with the Hamiltonian $H={\sum }_{n=0}^{N-1}\ln s{h}^{-2}(p_n/2)+\ln (\wp (x_n-x_{n+1})-\wp (x_n+x_{n+1})),x_N=x_0,$ 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.

25 pages, LaTeX