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

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


It is shown that the celebrated Menelaus relation, Hirota–Miwa bilinear equation for KP hierarchy and Fay's trisecant formula similar to the WDVV equation are associativity conditions for structure

Matrix Kadomtsev—Petviashvili Hierarchy and Spin Generalization of Trigonometric Calogero—Moser Hierarchy

We consider solutions of the matrix Kadomtsev-Petviashvili (KP) hierarchy that are trigonometric functions of the first hierarchical time t 1 = x and establish the correspondence with the spin

Toda lattice hierarchy and soliton equations on square lattice

Lattice soliton equations of KdV-family and Boussinesq-family are investigated from the viewpoint of the Toda lattice hierarchy. As a consequence, special solutions such as soliton solutions and

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We considered the relation between two famous integrable equations: The Hirota difference equation (HDE) and the Darboux system that describes conjugate curvilinear systems of coordinates in R 3 . We

Time discretization of the spin Calogero-Moser model and the semi-discrete matrix KP hierarchy

  • A. Zabrodin
  • Mathematics
    Journal of Mathematical Physics
  • 2019
We introduce the discrete time version of the spin Calogero-Moser system. The equations of motion follow from the dynamics of poles of rational solutions to the matrix KP hierarchy with discrete

Higher Hirota Difference Equations and Their Reductions

  • A. Pogrebkov
  • Mathematics
    Theoretical and Mathematical Physics
  • 2018
We previously proposed an approach for constructing integrable equations based on the dynamics in associative algebras given by commutator relations. In the framework of this approach, evolution



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GL3-invariant, finite-dimensional solutions of the Yang-Baxter equations acting in the tensor product of two irreducible representations of the group GL3 are investigated. A number of relations are


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A nonlinear partial difference equation which reduces to the sine-Gordon equation in the continuum limit, is obtained and solved by the method of dependent variable transformation. N -soliton

Theta Functions on Riemann Surfaces

Riemann's theta function.- The prime-form.- Degenerate Riemann surfaces.- Cyclic unramified coverings.- Ramified double coverings.- Bordered Riemann surfaces.

Algebro-geometric approach to nonlinear integrable equations

A brief but self-contained exposition of the basics of Riemann surfaces and theta functions prepares the reader for the main subject of this text, namely the application of these theories to solving

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A review of selected topics in Hirota’s bilinear difference equation (HBDE) is given. This famous 3-dimensional difference equation is known to provide a canonical integrable discretization for most

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We discuss an interrelation between quantum integrable models and classical soliton equations with discretized time. It appeared that spectral characteristics of quantum integrable systems may be