• DOI:10.1007/s002200050333
• Corpus ID: 204929063

Elliptic solutions to difference non-linear equations and related many-body problems

@inproceedings{IKrichever1997EllipticST,
  title={Elliptic solutions to difference non-linear equations and related many-body problems},
  author={I.Krichever and P.Wiegmann and A.Zabrodin},
  year={1997}
}

• I.Krichever, P.Wiegmann, A.Zabrodin
• Published 11 April 1997
• Mathematics

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.