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

Abstract

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 $\tau$-functions. Starting from a given algebraic curve, we express the $\tau$-function and the Baker-Akhiezer function in terms of the Riemann theta function. We show that the elliptic solutions, when the $\tau$-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.

22 pages, Latex with emlines2.sty

References in corpus (4)

• Quantum Integrable Systems and Elliptic Solutions of Classical Discrete Nonlinear Equations
• Spin generalization of the Ruijsenaars-Schneider model, non-abelian 2D Toda chain and representations of Sklyanin algebra
• A survey of Hirota's difference equations
• Bethe Ansatz and Classical Hirota Equation

Cited by in corpus (23)

• Integrable discrete Schrodinger equations and a characterization of Prym varieties by a pair of quadrisecants
• Antiperiodic dynamical 6-vertex model by separation of variables II: Functional equations and form factors
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• Elliptic solutions to integrable nonlinear equations and many-body systems
• Desargues maps and the Hirota-Miwa equation
• The Hirota equation over finite fields. Algebro-geometric approach and multisoliton solutions
• Bilinearization and Casorati determinant solution to the non-autonomous discrete KdV equation
• Bilinear recurrences and addition formulae for hyperelliptic sigma functions
• Field analogue of the Ruijsenaars-Schneider model
• On Elliptic Lax Systems on the Lattice and a Compound Theorem for Hyperdeterminants
• Hirota difference equation: IST, Darboux transformation and solitons
• Integrable 1D Toda cellular automata
• Symmetries of the Hirota Difference Equation
• Darboux transformations for linear operators on two dimensional regular lattices
• Hirota equation and the quantum plane
• Theta function parameterization and fusion for 3-D integrable Boltzmann weights
• Nested Bethe Ansatz and Finite Dimensional Canonical Commutation Relations
• Time discretization of the spin Calogero-Moser model and the semi-discrete matrix KP hierarchy
• The modified tetrahedron equation and its solutions
• Finite-genus solutions for the Hirota's bilinear difference equation
• Solitons on Noncommutative Torus as Elliptic Calogero Gaudin Models, Branes and Laughlin Wave Functions
• Multiple addition theorem for discrete and continuous nonlinear problems
• Menelaus relation and Fay's trisecant formula are associativity equations