Volterra Chain Research Articles

Commutative Poisson algebras from deformations of noncommutative algebras

It is well-known that a formal deformation of a commutative algebra A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document} leads to a Poisson bracket on A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document} and that the classical limit of a derivation on the deformation leads to a derivation on A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}, which is Hamiltonian with respect to the Poisson bracket. In this paper we present a generalization of it for formal deformations of an arbitrary noncommutative algebra A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}. The deformation leads in this case to a Poisson algebra structure on Π(A):=Z(A)×(A/Z(A))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi (\\mathcal {A}){:}{=}Z(\\mathcal {A})\ imes (\\mathcal {A}/Z(\\mathcal {A}))$$\\end{document} and to the structure of a Π(A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi (\\mathcal {A})$$\\end{document}-Poisson module on A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}. The limiting derivations are then still derivations of A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}, but with the Hamiltonian belong to Π(A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi (\\mathcal {A})$$\\end{document}, rather than to A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}. We illustrate our construction with several cases of formal deformations, coming from known quantum algebras, such as the ones associated with the nonabelian Volterra chains, Kontsevich integrable map, the quantum plane and the quantized Grassmann algebra.

ОСОБЕННОСТИ ПРИМЕНЕНИЯ МЕТОДОВ СТАТИСТИЧЕСКОЙ МЕХАНИКИ ПРИ ИССЛЕДОВАНИИ ОБОБЩЕНИЙ КЛАССИЧЕСКОЙ ЦЕПОЧКИ ВОЛЬТЕРРА

An analytical study of one of the generalizations of the classical Volterra chain (coupled Volterra chain) has been carried out. The use of methods and approaches of statistical mechanics in the study made it possible to identify a number of important features of such models, which are currently widely used in research in chemistry, ecology, biology, economics, hydrodynamics, astrophysics, plasma physics. The advantage of the studied model is that the introduction of additional variables makes it possible to take into account a greater number of factors influencing the dynamics of the system without changing its algebraic structure. However, an attempt at a statistical description of the model reveals a significant drawback - the partition function of the system becomes divergent. In this regard, the possibility of an adequate statistical description of the coupled Volterra chain in the case of divergence of its partition function is investigated. For the selected specific model, the sources of the divergence of the partition function are identified, and a method for their cancel out is proposed. The partition function is calculated in an explicit and finite form, which allows us to state that an adequate statistical description of the phenomena described by this model is possible. Given the wide range of applications of the coupled Volterra chain, the results obtained are of interest to theorists and practitioners when analyzing and predicting the results of a statistical study of various generalizations of the model.

Waves in strongly nonlinear Gardner-like equations on a lattice

We introduce and study a family of lattice equations which may be viewed either as a strongly nonlinear discrete extension of the Gardner equation, or a non-convex variant of the Lotka–Volterra chain. Their deceptively simple form supports a very rich family of complex solitary patterns. Some of these patterns are also found in the quasi-continuum rendition, but the more intriguing ones, like interlaced pairs of solitary waves, or waves which may reverse their direction either spontaneously or due a collision, are an intrinsic feature of the discrete realm.

Invariant manifolds and separation of the variables for integrable chains

A notion of the generalized invariant manifold for a nonlinear integrable lattice is considered. Earlier, it has been observed that this kind of objects provide an effective tool for evaluating the recursion operators and Lax pairs. In this article we show with an example of the Volterra chain that the generalized invariant manifold can be used for constructing exact particular solutions as well. To this end we first find an invariant manifold depending on two constant parameters. Then we assume that an ordinary difference equation (ODE) defining the generalized invariant manifold has a solution polynomially depending on one of the spectral parameters and derive an ODE, for the roots of the polynomials. The efficiency of the method is approved by some illustrative examples.

Statistical Description of Closed Biocenoses Described by a Volterra Chain Subject to Periodic Boundary Conditions

Statistical Description of Closed Biocenoses Described by a Volterra Chain Subject to Periodic Boundary Conditions

Законы сохранения в задаче о ступеньке для цепочки Вольтерра

Законы сохранения в задаче о ступеньке для цепочки Вольтерра

Volterra Chain and Catalan Numbers

We consider the Cauchy problem for the Volterra chain with an initial condition equal to 0 in one node and 1 in the others. It is shown that this problem admits an exact solution in terms of the Bessel functions. The Taylor series arising here are related to the exponential generating function for Catalan numbers.

Poisson Λ-brackets for Differential–Difference Equations

Abstract We introduce the notion of a multiplicative Poisson $\lambda$-bracket, which plays the same role in the theory of Hamiltonian differential–difference equations as the usual Poisson $\lambda$-bracket plays in the theory of Hamiltonian partial differential equations (PDE). We classify multiplicative Poisson $\lambda$-brackets in one difference variable up to order 5. As an example, we demonstrate how to apply the Lenard–Magri scheme to a compatible pair of multiplicative Poisson $\lambda$-brackets of order 1 and 2, to establish integrability of the Volterra chain.

On a method for constructing the Lax pairs for integrable models via a quadratic ansatz

A method for constructing the Lax pairs for nonlinear integrable models is suggested. First we look for a nonlinear invariant manifold of the linearization of the given equation. Examples show that such an invariant manifold does exist and can effectively be found. Actually, it is defined by a quadratic form. As a result we get a nonlinear Lax pair consisting of the linearized equation and the invariant manifold. Our second step consists of finding an appropriate change of the variables to linearize the found nonlinear Lax pair. The desired change of the variables is again defined by a quadratic form. The method is illustrated by the well-known KdV equation, the modified Volterra chain and a less studied coupled lattice connected to the affine Lie algebra . New Lax pairs are found. The formal asymptotic expansions for their eigenfunctions are constructed around the singular values of the spectral parameter. By applying the method of the formal diagonalization to these Lax pairs, the infinite series of the local conservation laws are obtained for the corresponding nonlinear models.

On the Construction of Elliptic Solutions of Integrable Birational Maps

ABSTRACTWe present a systematic technique to find explicit solutions of birational maps, provided that these solutions are given in terms of elliptic functions. The two main ingredients are the following: (i) application of classical addition theorems for elliptic functions and (ii) experimental technique to detect an algebraic curve containing a given sequence of points in a plane. These methods are applied to Kahan–Hirota–Kimura discretizations of the periodic Volterra chains with three and four particles.

On solitary patterns in Lotka–Volterra chains

We present and study a class of Lotka–Volterra chains with symmetric -neighbors interactions. To identify the types of solitary waves which may propagate along the chain, we study their quasi-continuum approximations which, depending on the coupling between neighbors, reduce into a large variety of partial differential equations. Notable among the emerging equations is a bi-cubic equation which we study in some detail. It begets remarkably stable topological and non-topological solitary compactons that interact almost elastically. They are used to identify discretons, their solitary discrete antecedents on the lattice, which decay at a doubly exponential rate. Many of the discrete modes are robust while others either decompose or evolve into breathers.

Representations of sl(2, ℂ) in category O and master symmetries

In this paper, we first give a short account on the indecomposable sl(2,C) modules in the Bernstein-Gelfand-Gelfand (BGG) category O. We show these modules naturally arise for homogeneous integrable nonlinear evolutionary systems. We then develop an approach to construct master symmetries for such integrable systems. This method naturally enables us to compute the hierarchy of time-dependent symmetries. We finally illustrate the method using both classical and new examples. We compare our approach to the known existing methods used to construct master symmetries. For the new integrable equations such as a Benjamin-Ono type equation, a new integrable Davey-Stewartson type equation and two different versions of (2+1)-dimensional generalised Volterra Chains, we generate their conserved densities using their master symmetries.

Explicit solutions for a (2 + 1)-dimensional Toda-like chain

We consider a (2 + 1)-dimensional Toda-like chain which can be viewed as a two-dimensional generalization of the Wu–Geng model and which is closely related to the two-dimensional Volterra, two-dimensional Toda and relativistic Toda lattices. In the framework of the Hirota direct approach, we present equations describing this model as a system of bilinear equations that belongs to the Ablowitz–Ladik hierarchy. Using the Jacobi-like determinantal identities and the Fay identity for the theta-functions, we derive its Toeplitz, dark-soliton and quasiperiodic solutions as well as the similar set of solutions for the two-dimensional Volterra chain.

KdV–Volterra chain

This paper is devoted to the system of coupled KdV-like equations. It is shown that this apparently non-integrable system possesses an integrable reduction which is closely related to the Volterra chain. This fact is used to construct the hyperelliptic solutions of the original system.

The Cauchy problem for a semi-infinite Volterra chain with an asymptotically periodic initial condition

We examine the Cauchy problem for a semi-infinite Volterra chain with an asymptotically periodic initial condition. The question is addressed of existence of a solution with the same asymptotics at infinity as the initial condition. We demonstrate that the method of the inverse scattering problem is applicable to this problem.

Lenard scheme for two-dimensional periodic Volterra chain

We prove that for compatible weakly nonlocal Hamiltonian and symplectic operators, hierarchies of infinitely many commuting local symmetries and conservation laws can be generated under some easily verified conditions no matter whether the generating Nijenhuis operators are weakly nonlocal or not. We construct a recursion operator of the two-dimensional periodic Volterra chain from its Lax representation and prove that it is a Nijenhuis operator. Furthermore we show that this system is a (generalized) bi-Hamiltonian system. Rather surprisingly, the product of its weakly nonlocal Hamiltonian and symplectic operators gives rise to the square of the recursion operator.

Integrable boundary-value problem for the Volterra chain on the half-axis

We study quasiperiodic (finite-gap) solutions of the Volterra chain satisfying an integrable boundary condition on the semiaxis. From the set of general finite-gap solutions, only those corresponding to the boundary-value problem are singled out, the relevant condition being expressed as a system of algebraic equations.

Invariant Submanifolds of the Darboux–Kadomtsev–Petviashvili Chain and an Extension of the Discrete Kadomtsev–Petviashvili Hierarchy

We investigate invariant submanifolds of the so-called Darboux–Kadomtsev–Petviashvili chain. We show that restricting the dynamics to a class of invariant submanifolds yields an extension of the discrete Kadomtsev–Petviashvili hierarchy and intersections of invariant submanifolds yield the Lax description of a wide class of differential–difference systems. We consider self-similar reductions. We show that self-similar substitutions result in purely discrete equations that depend on a finite set of parameters and in equations determining deformations w.r.t. these parameters. We present examples. In particular, we show that the well-known first discrete Painleve equation corresponds to the Volterra chain hierarchy. We derive the equations naturally generalizing the first discrete Painleve equation in the sense that all of them become the first Painleve equation in the continuum limit.

Reductions of the Volterra and Toda chains

We consider Volterra and Toda systems. Using certain algebraic relations, we construct a denumerable class of reductions of these differential-difference systems. The result is given by finite-dimensional systems with Bäcklund autotransformations.

A general approach to the modelling of trophic chains

Based on the law of mass action (and its microscopic foundation) and mass conservation, we present here a method to derive consistent dynamic models for the time evolution of systems with an arbitrary number of species. Equations are derived through a mechanistic description, ensuring that all parameters have ecological meaning. After discussing the biological mechanisms associated to the logistic and Lotka–Volterra equations, we show how to derive general models for trophic chains, including the effects of internal states at fast time scales. We show that conformity with the mass action law leads to different functional forms for the Lotka–Volterra and trophic chain models. We use mass conservation to recover the concept of carrying capacity for an arbitrary food chain.