Communications in Mathematical Physics

, Volume 193, Issue 2, pp 373–396 | Cite as

Elliptic Solutions to Difference Non-Linear Equations and Related Many-Body Problems

  • I. Krichever
  • P. Wiegmann
  • A. Zabrodin


We study algebro-geometric (finite-gap) and elliptic solutions of fully discretized KP or 2D Toda equations. In bilinear form they are Hirota's difference equation for τ-functions. Starting from a given algebraic curve, we express the τ-function and the Baker–Akhiezer function in terms of the Riemann theta function. We show that the elliptic solutions, when the τ-function is an elliptic polynomial, form a subclass of the general algebro-geometric solutions. We construct the algebraic curves of the elliptic solutions. The evolution of zeros of the elliptic solutions is governed by the discrete time generalization of the Ruijsenaars-Schneider many body system. The zeros obey equations which have the form of nested Bethe-ansatz equations, known from integrable quantum field theories. We discuss the Lax representation and the action-angle-type variables for the many body system. We also discuss elliptic solutions to discrete analogues of KdV, sine-Gordon and 1D Toda equations and describe the loci of the zeros.


Theta Function Body System Algebraic Curve Discrete Analogue Time Generalization 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • I. Krichever
    • 1
  • P. Wiegmann
    • 2
  • A. Zabrodin
    • 3
  1. 1.Department of Mathematics of Columbia University and Landau Institute for Theoretical Physics, Kosygina str. 2, 117940 Moscow, RussiaRU
  2. 2.James Franck Institute and Enrico Fermi Institute of the University of Chicago, 5640 S.Ellis Avenue, Chicago, IL 60637, USA and Landau Institute for Theoretical PhysicsUS
  3. 3.Joint Institute of Chemical Physics, Kosygina str. 4, 117334, Moscow, Russia and ITEP, 117259, Moscow, RussiaRU

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