Volterra Lattice Research Articles

An approach to directly construct exact solutions of nonlinear differential-difference equations

In this paper, a constructive method for exactly solving nonlinear differential-difference equations (NDDEs) is presented. The NDDE which includes Hybrid lattice, discretized mKdV lattice and modified Volterra lattice is chosen to illustrate this approach. Some new solutions of these lattices are obtained.

Reductions of the Volterra lattice

We exhibit three classes of algebraic constraints which are shown compatible with Volterra lattice.

Gel'fand-Zakharevich systems and algebraic integrability: the Volterra lattice revisited

In this paper we will discuss some features of the bi-Hamiltonian method for solving the Hamilton-Jacobi (H-J) equations by Separation of Variables, and make contact with the theory of Algebraic Complete Integrability and, specifically, with the Veselov–Novikov notion of algebro-geometric (AG) Poisson brackets. The bi-Hamiltonian method for separating the Hamilton-Jacobi equations is based on the notion of pencil of Poisson brackets and on the Gel’fand-Zakharevich (GZ) approach to integrable systems. We will herewith show how, quite naturally, GZ systems may give rise to AG Poisson brackets, together with specific recipes to solve the H-J equations. We will then show how this setting works by framing results by Veselov and Pensko¨o about the algebraic integrability of the Volterra lattice within the bi-Hamiltonian setting for Separation of Variables

Non-isospectral lattice hierarchies in 2+1 dimensions and generalized discrete Painlevé hierarchies

In a recent paper we introduced a new 2 + 1-dimensional non-isospectral extension of the Volterra lattice hierarchy, along with its corresponding hierarchy of underlying linear problems. Here we consider reductions of this lattice hierarchy to hierarchies of discrete equations, which we obtain once again along with their hierarchy of underlying linear problems. We obtain a generalized discrete first Painlevé hierarchy which includes as special cases, after further summation, both the standard discrete first Painlevé hierarchy and a new extended version of the discrete thirty-fourth Painlevé hierarchy.

Multi-Component Generalizations of Four Integrable Differential-Difference Equations: Soliton Solutions and Bilinear Bäcklund Transformations

Bilinear approach is applied to derive integrable multi-component generalizations of the so-called 1+1 dimensional special Toda lattice, the Volterra lattice, a simple differential-difference equation found by Adler, Moser, Weiss, Veselov and Shabat and another integrable lattice reduced from the discrete BKP equation. Their soliton solutions expressed by pfaffians and the corresponding bilinear Backlund transformations are obtained.

Bogoyavlensky–Volterra and Birkhoff integrable systems

In this paper we examine an interesting connection between the generalized Volterra lattices of Bogoyavlensky and a special case of an integrable system defined by Sklyanin. The Sklyanin system happens to be one of the cases in the classification of Kozlov and Treshchev of Birkhoff integrable Hamiltonian systems. Using this connection we demonstrate the integrability of the system and define a new Lax pair representation. In addition, we comment on the bi-Hamiltonian structure of the system.

Surface diffusion on a square lattice with two non-equivalent sites

We investigate surface diffusion in a system of particles adsorbed on a square lattice with two non-equivalent sites. Analytical expression for the chemical diffusion coefficient have been derived in case of strong inhomogeneity of the lattice. It is shown that the character of the particle migration depends substantially on the relation between the jump frequencies. When the frequencies differ insignificantly, the particle diffusion proceeds by single uncorrelated jumps of particles between the nearest neighbor lattice sites. In the opposite case, when the jump frequencies differ considerably (slow and fast jumps), the transfer of the particles over the surface proceeds by the pairs of consecutive jumps: a slow jump is followed by a fast jump. There are two types of such jump pairs. We have calculated the coverage dependencies of the tracer, jump and chemical diffusion coefficients for the Langmuir lattice gas for some representative temperatures using analytical and numerical approaches. The coincidence of the data obtained by the two quite different methods is rather good.

The matrix realization of affine Jacobi varieties and the extended Lotka–Volterra lattice

We study completely integrable Hamiltonian systems whose monodromy matrices are related to the representatives for the set of gauge equivalence classes $\boldsymbol{\mathcal{M}}_F$ of polynomial matrices. Let $X$ be the algebraic curve given by the common characteristic equation for $\boldsymbol{\mathcal{M}}_F$. We construct the isomorphism from the set of representatives to an affine part of the Jacobi variety of $X$. This variety corresponds to the invariant manifold of the system, where the Hamiltonian flow is linearized. As the application, we discuss the algebraic completely integrability of the extended Lotka-Volterra lattice with a periodic boundary condition.

Q4: integrable master equation related to an elliptic curve

One of the most fascinating and technically demanding parts of the theory of two-dimensional integrable systems constitutes the models with the spectral parameter on an elliptic curve, including Landau-Lifshitz and Krichever-Novikov equations as well as elliptic Toda and Ruijsenaars-Toda lattices, elliptic Volterra lattice and Shabat-Yamilov lattice. We explain how all these models can be unified on the basis of a single equation Q4 on a quad-graph. This discrete model plays the role of the “master equation” in the (sl2 part of) 2D integrability: most of other models in this area can be obtained from this one by certain limiting procedures. More precisely, discrete versions of all of the above-mentioned models appear from Q4 imposed on a multidimensional cubic lattice. Zero-curvature representations for all models appear as a byproduct of the general construction.

Computation of densities and fluxes of nonlinear differential‐difference equations

Direct methods to find conserved densities and fluxes of differentialdifference equations are presented and illustrated for the Kacvan Moerbeke (KvM) and modified Volterra lattices. A Miura map whi...

Lax pairs and conservation laws for two differential-difference systems

A coupled extended Lotka–Volterra lattice and a special Toda lattice are derived from the existing bilinear equations. Starting from the corresponding bilinear Bäcklund transformation, Lax pairs for these two differential-difference systems are obtained. Furthermore, an infinite number of conservation laws for the differential-difference equations are deduced from the Lax pairs in a systematic way.

The extended Lotka–Volterra lattice and affine Jacobi varieties of spectral curves

Based on the work by Smirnov and Zeitlin, we study a simple realization of the matrix construction of the affine Jacobi varieties. We find that the realization is given by a classical integrable model, the extended Lotka–Volterra lattice. We investigate the integrable structure of the representative for the gauge equivalence class of matrices, which is isomorphic to the affine Jacobi variety, and make use of it to discuss the solvability of the model.

From the Toda lattice to the Volterra lattice and back

We discuss the relationship between the multiple Hamiltonian structures of the generalized Toda lattices and that of the generalized Volterra lattices.

Infinitely many conservation laws and integrable discretizations for some lattice soliton equations

In this paper, by means of the Lax representations, we demonstrate the existence of infinitely many conservation laws for the general Toda-type lattice equation, the relativistic Volterra lattice equation, the Suris lattice equation and some other lattice equations. The conserved density and the associated flux are given formulaically. We also give an integrable discretization for a lattice equation with n dependent coefficients.

A Nonlocal Kac-van Moerbeke Equation Admitting N-Soliton Solutions

Using our previous work on reflectionless analytic difference operators and a nonlocal Toda equation, we introduce analytic versions of the Volterra and Kac-van Moerbeke lattice equations. The real-valued N-soliton solutions to our nonlocal equations correspond to self-adjoint reflectionless analytic difference operators with N bound states. A suitable scaling limit gives rise to the N-soliton solutions of the Korteweg-de Vries equation.

Integrable hierarchies of nonlinear difference-difference equations and symmetries

We construct the hierarchy of nonlinear difference-difference equations associated with the discrete Schrodinger spectral problem. As examples of equations contained in this hierarchy we obtain the discrete-time Toda and Volterra lattice equations. In the case of the time-discrete Toda lattice, we construct its Lie point and generalized symmetries. Finally, we present its Backlund transformations and relate it to the already constructed symmetries.

Hyperelliptic Prym Varieties and Integrable Systems

We introduce two algebraic completely integrable analogues of the Mumford systems which we call hyperelliptic Prym systems, because every hyperelliptic Prym variety appears as a fiber of their momentum map. As an application we show that the general fiber of the momentum map of the periodic Volterra lattice is an affine part of a hyperelliptic Prym variety, obtained by removing n translates of the theta divisor, and we conclude that this integrable system is algebraic completely integrable.

On Integrable Discretization of the Inhomogeneous Volterra Lattice

An integrable discretization of an inhomogeneous Volterra lattice is introduced. Some aspects of dynamics of a soliton affected by a random force are discussed.

On Integrable Discretization of the Inhomogeneous Volterra Lattice

An integrable discretization of an inhomogeneous Volterra lattice is introduced. Some aspects of dynamics of a soliton affected by a random force are discussed.

Bilinear structure and determinant solution for the relativistic Lotka–Volterra equation

The relativistic Lotka–Volterra (RLV) lattice and the discrete-time relativistic Lotka–Volterra (dRLV) lattice are investigated by using the bilinear formalism. The bilinear equations for them are systematically constructed with the aid of the singularity confinement test. It is shown that the RLV lattice and dRLV lattice are decomposed into the Bäcklund transformations of the Toda lattice system. The N-soliton solutions are explicitly constructed in the form of the Casorati determinant.