1997/04/11 by I. M. Krichever, I. Krichever, P. Wiegmann +1 · 62 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic curve #Algebraic structures and combinatorial models #Bethe ansatz #Elliptic curve #Elliptic function #Elliptic rational functions #Integrable system #Jacobi elliptic functions #Korteweg–de Vries equation #Mathematical analysis #Mathematics #Nonlinear Waves and Solitons #Nonlinear system #Physics #Pure mathematics #Quarter period #Supersingular elliptic curve #Theta function #hep-th
paper · pdf · doi:10.1007/s002200050333
published in Communications in Mathematical Physics 193(2), 373-396 (Springer Science+Business Media) · 22 pages, Latex with emlines2.sty
arxiv created 1997/04/11 · openalex publication_date 1998/04/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.