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

I. Krichever, P. Wiegmann, A. Zabrodin

Research output: Contribution to journalArticlepeer-review

41 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)373-396
Number of pages24
JournalCommunications in Mathematical Physics
Issue number2
Publication statusPublished - 3 Apr 1998
Externally publishedYes


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