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

@inproceedings{IKrichever1997EllipticST, title={Elliptic solutions to difference non-linear equations and related many-body problems}, author={I.Krichever and P.Wiegmann and A.Zabrodin}, year={1997} }

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…

## 6 Citations

### MENELAUS RELATION, HIROTA–MIWA EQUATION AND FAY'S TRISECANT FORMULA ARE ASSOCIATIVITY EQUATIONS

- MathematicsJournal of Nonlinear Mathematical Physics
- 2021

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

- Mathematics
- 2020

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

- Mathematics, Physics
- 2020

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…

### Hirota Difference Equation and Darboux System: Mutual Symmetry

- MathematicsSymmetry
- 2019

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

- MathematicsJournal 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

- MathematicsTheoretical 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…

## References

SHOWING 1-10 OF 23 REFERENCES

### GL3-invariant solutions of the Yang-Baxter equation and associated quantum systems

- Mathematics, Physics
- 1986

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…

### Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles

- Mathematics
- 1980

### Quantum inverse scattering method on a spacetime lattice

- Physics
- 1992

The formalism of the quantum inverse scattering method is developed on a lattice in spacetime that imitates light cone coordinates. The zero-curvature representation makes it possible to introduce in…

### Nonlinear Partial Difference Equations III; Discrete Sine-Gordon Equation

- Mathematics, Physics
- 1977

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

- Mathematics
- 1973

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

- Mathematics, Physics
- 1994

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…

### A survey of Hirota's difference equations

- Mathematics
- 1997

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…

### Bethe ansatz and classical Hirota equation

- Mathematics, Physics
- 1996

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…