Elliptic analog of the Toda lattice

1999/09/30 by I. M. Krichever, Krichever, I. M. · 3 citations
Physics and Astronomy · #Black Holes and Theoretical Physics #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems #hep-th

paper · pdf · doi:10.48550/arxiv.hep-th/9909224

25 pages, LaTeX

arxiv created 1999/11/04 · arxiv updated 2009/11/30

Abstract

The action-angle variables for N-particle Hamiltonian system with the Hamiltonian H=∑_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=x₀, are constructed, and the system is solved in terms of the Riemann θ-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 w²-P_N^el(z)w+Λ^2N=0, where P_N^el(z) is an elliptic function with pole of order N at the point z=0.

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• Elliptic Ruijsenaars-Toda and elliptic Toda chains: classical r-matrix structure and relation to XYZ chain
• Seiberg-Witten Theory, Symplectic Forms, and Hamiltonian Theory of Solitons
• Classical r-matrix structure for elliptic Ruijsenaars chain and 1+1 field analogue of Ruijsenaars-Schneider model

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