TY - JOUR

T1 - Elliptic solutions to difference non-linear equations and related many-body problems

AU - Krichever, I.

AU - Wiegmann, P.

AU - Zabrodin, A.

PY - 1998/4/3

Y1 - 1998/4/3

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0032478429&partnerID=8YFLogxK

U2 - 10.1007/s002200050333

DO - 10.1007/s002200050333

M3 - Article

AN - SCOPUS:0032478429

VL - 193

SP - 373

EP - 396

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -