KdV-Volterra chain

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 this system is a (generalised) bi-Hamiltonian system. Rather surprisingly, the product of its weakly nonlocal Hamiltonian and sym…

The Generalized Dirichlet to Neumann map for the KdV equation on the half-line

For the two versions of the KdV equation on the positive half-line an initial-boundary value problem is well posed if one prescribes an initial condition plus either one boundary condition if $q_{t}$ and $q_{xxx}$ have the same sign (KdVI) or two boundary conditions if $q_{t}$ and $q_{xxx}$ have opposite sign (KdVII). Constructing the generalized Dirichlet to Neumann map for the above problems means characterizing the unknown boundary values in terms of the given initial and boundary conditions. For example, if $\{q(x,0),q(0,t) \}$ and $\{q(x,0),q(0,t),q_{x}(0,t) \}$ are given for the KdVI and…

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.

Direct Connection between the R_{II} Chain and the Nonautonomous Discrete Modified KdV Lattice

The spectral transformation technique for symmetric R_{II} polynomials is developed. Use of this technique reveals that the nonautonomous discrete modified KdV (nd-mKdV) lattice is directly connected with the R_{II} chain. Hankel determinant solutions to the semi-infinite nd-mKdV lattice are also presented.

Integrable reductions of the dressing chain

In this paper we construct a family of integrable reductions of the dressing chain, described in its Lotka-Volterra form. For each $k,n\in\mathbb N$ with $n\geqslant 2k+1$ we obtain a Lotka-Volterra system $\hbox{LV}_b(n,k)$ on $\mathbb R^n$ which is a deformation of the Lotka-Volterra system $\hbox{LV}(n,k)$, which is itself an integrable reduction of the $2m+1$-dimensional Bogoyavlenskij-Itoh system $\hbox{LV}(2m+1,m)$, where $m=n-k-1$. We prove that $\hbox{LV}_b(n,k)$ is both Liouville and non-commutative integrable, with rational first integrals which are deformations of the rational first…

A geometric derivation of KdV-type hierarchies from root systems

For the root system of each complex semi-simple Lie algebra of rank two, and for the associated 2D Toda chain $E=\{u_{xy}=\exp(K u)\}$, we calculate the two first integrals of the characteristic equation $D_y(w)=0$ on $E$. Using the integrals, we reconstruct and make coordinate-independent the $(2\times 2)$-matrix operators $\square$ in total derivatives that factor symmetries of the chains. Writing other factorizations that involve the operators $\square$, we obtain pairs of compatible Hamiltonian operators that produce KdV-type hierarchies of symmetries for $\cE$. Having thus reduced the pro…

非交换离散方程、对称性与约化

Noncommutative discrete equations, symmetries and reductions

本文利用非交换离散KdV方程的Lax对构造广义对称与Miura变换，通过对称约化得到非交换离散Painlevé方程和推广的Ernst方程，并建立Darboux变换、自Bäcklund变换与非交换Yang–Baxter映射的联系。

Dressing the Dressing Chain

The dressing chain is derived by applying Darboux transformations to the spectral problem of the Korteweg-de Vries (KdV) equation. It is also an auto-Bäcklund transformation for the modified KdV equation. We show that by applying Darboux transformations to the spectral problem of the dressing chain one obtains the lattice KdV equation as the dressing chain of the dressing chain and, that the lattice KdV equation also arises as an auto-Bäcklund transformation for a modified dressing chain. In analogy to the results obtained for the dressing chain (Veselov and Shabat proved complete integrabilit…

Boundary Value Problems For Integrable Equations Compatible With The Symmetry Algebra

Boundary value problems for integrable nonlinear partial differential equations are considered from the symmetry point of view. Families of boundary conditions compatible with the Harry-Dym, KdV and MKdV equations and the Volterra chain are discussed. We also discuss the uniqueness of some of these boundary conditions.

Lax pair formulation of the N=4 Toda chain (KdV) hierarchy in N=4 superspace

Lax pair and Hamiltonian formulations of the N=4 supersymmetric Toda chain (KdV) hierarchy in N=4 superspace are proposed. The general formulae for the infinite tower of its bosonic flows in terms of the Lax operator in N=4 superspace are derived. A new N=4 superfield basis in which the flows are local is constructed and its embedding in the N=4 O(4) superconformal supercurrent is established. A proof that the flows possess five complex conjugations and aninfinite-dimensional group of discrete symmetries in N=4 superspace is presented. A relation between the two descriptions of the hierarchy i…

Toda and Volterra Lattice Equations from Discrete Symmetries of KP Hierarchies

The discrete models of the Toda and Volterra chains are being constructed out of the continuum two-boson KP hierarchies. The main tool is the discrete symmetry preserving the Hamiltonian structure of the continuum models. The two-boson currents of KP hierarchy are being associated with sites of the corresponding chain by successive actions of discrete symmetry.

Korteweg-de Vries and Fermi-Pasta-Ulam-Tsingou: asymptotic integrability of quasi unidirectional waves

In this paper we construct a higher order expansion of the manifold of quasi unidirectional waves in the Fermi-Pasta-Ulam (FPU) chain. We also approximate the dynamics on this manifold. As perturbation parameter we use $h^2=1/n^2$, where $n$ is the number of particles of the chain. It is well known that the dynamics of quasi unidirectional waves is described to first order by the Korteweg-de Vries (KdV) equation. Here we show that the dynamics to second order is governed by a combination of the first two nontrivial equations in the KdV hierarchy -- for any choice of parameters in the FPU poten…

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 s…

Discrete Hirota reductions associated with the lattice KdV equation

We study the integrability of a family of birational maps obtained as reductions of the discrete Hirota equation, which are related to travelling wave solutions of the lattice KdV equation. In particular, for reductions corresponding to waves moving with rational speed N/M on the lattice, where N,M are coprime integers, we prove the Liouville integrability of the maps when N + M is odd, and prove various properties of the general case. There are two main ingredients to our construction: the cluster algebra associated with each of the Hirota bilinear equations, which provides invariant (pre)sym…

Integrable Discrete Nonautonomous Quad-equations as Bäcklund Auto-transformations for Known Volterra and Toda Type Semidiscrete Equations

We construct integrable discrete nonautonomous quad-equations as Bäcklund auto-transformations for known Volterra and Toda type semidiscrete equations, some of which are also nonautonomous. Additional examples of this kind are found by using transformations of discrete equations which are invertible on their solutions. In this way we obtain integrable examples of different types: discrete analogs of the sine-Gordon equation, the Liouville equation and the dressing chain of Shabat. For Liouville type equations we construct general solutions, using a specific linearization. For sine-Gordon type …

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

A method for constructing the Lax pairs for nonlinear integrable models is suggested. First we look for a nonlinear invariant manifold to the linearization of the given equation. Examples show that such 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 f…

可积微分差分方程的有限维约化

Finite dimensional reductions of integrable differential-difference equations

本文提出约化理想概念，为可积微分差分方程引入新约化类，产生交换与非交换环境下的有限维可积系统，并以Volterra层次为例说明。

通过三维穿衣链构造非线性偏微分方程的精确解

Construction of exact solutions of nonlinear PDE via dressing chain in 3D

本文研究Davey-Stewartson型耦合系统与二维Toda型晶格的对偶性，提出新耦合系统、消除非局域性的有限约化方法及构造显式解的算法，并给出与Volterra晶格相关的新解。

Differential equations for the closed geometric crystal chains

We present two types of systems of differential equations that can be derived from a set of discrete integrable systems which we call the closed geometric crystal chains. One is a kind of extended Lotka-Volterra systems, and the other seems to be generally new but reduces to a previously known system in a special case. Both equations have Lax representations associated with what are known as the loop elementary symmetric functions, which were originally introduced to describe products of affine type A geometric crystals for symmetric tensor representations. Examples of the derivations of the c…

Invariant Submanifolds of Darboux-KP Chain and Extension of the Discete KP Hierarchy

Invariant submanifolds of the so-called Darboux-KP chain are investigated. It is shown that restriction of dynamics on some class of invariant submanifolds yields the extension of the discrete KP hierarchy, while the intersections leads to Lax pairs for a broad class of differential-difference systems with finite number of fields. Some attention is given to investigation of self-similar reductions. It is shown that self-similar ansatzes lead to purely discrete equations with dependence on some number of parameters together with equations governing deformations with respect to these parameters.…

Cnoidal Waves on Fermi-Pasta-Ulam Lattices

We study a chain of infinitely many particles coupled by nonlinear springs, obeying the equations of motion [\ddot{q}_n = V'(q_{n+1}-q_n) - V'(q_n-q_{n-1})] with generic nearest-neighbour potential $V$. We show that this chain carries exact spatially periodic travelling waves whose profile is asymptotic, in a small-amlitude long-wave regime, to the KdV cnoidal waves. The discrete waves have three interesting features: (1) being exact travelling waves they keep their shape for infinite time, rather than just up to a timescale of order wavelength$^{-3}$ suggested by formal asymptotic analysis, (…

Toda型三维格点的对称性

Symmetries of Toda type 3D lattices

该研究探讨Davey-Stewartson型耦合系统与二维Toda型格点的对偶性，描述新可积格点的对称性层级并给出显式对称形式，提出基于修饰链Darboux可积约化构造精确解的方法，还给出Volterra格点相关耦合系统的新解示例。

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 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 ordinary difference equation defining the generalized invariant manifold has a solution polynomial…

Representations of sl(2,C) in the BGG 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 equatio…

Semi-discrete Lagrangian 2-forms and the Toda hierarchy

We present a variational theory of integrable differential-difference equations (semi-discrete integrable systems). This is a natural extension of the ideas known by the names "Lagrangian multiforms" and "Pluri-Lagrangian systems", which have previously been established in both the fully discrete and fully continuous cases. The main feature of these ideas is to capture a hierarchy of commuting equations in a single variational principle. Our main example to illustrate the new semi-discrete theory of Lagrangian multiforms is the Toda lattice. This ODE describes the evolution in continuous time …

随机矩阵族与可积微分恒等式

Random matrix ensembles and integrable differential identities

本文研究了随机矩阵理论中单位和正交族的可积微分恒等式，通过构造连续极限下的动力学链，揭示了热力学极限下的可积系统特性。