Symmetry groups of differential-difference equations and their compatibility

It is shown that the intrinsic determining equations of a given differential-difference equation (DDE) can be derived by the compatibility between the original equation and the intrinsic invariant surface condition. The (2+1)-dimensional Toda lattice, the special Toda lattice and the DD-KP equation serving as examples are used to illustrate this approach. Then, Bäcklund transformations of the (2+1)-dimensional DDEs including the special Toda lattice, the modified Toda lattice and the DD-KZ equation are presented by using the non-intrinsic direct method. In addition, the Clarkson-Kruskal direct method is developed to find similarity reductions of the DDEs.     

Rational formal solutions of differential-difference equations

In this paper, we study rational formal solutions of differential-difference equations by using a generalized ansätz. With the help of symbolic computation Maple, we obtain many explicit exact solutions of differential-difference equations(DDEs). The solutions contain solitary wave solutions and periodic wave solutions. The (2 + 1)-dimensional Toda lattice equation, relativistic Toda lattice equation and the discrete mKdV equation are chosen to illustrate our algorithm.     

Adomian decomposition method for nonlinear differential-difference equations

We extend Adomian decomposition method (ADM) to find the approximate solutions for the nonlinear differential-difference equations (NDDEs), such as the discretized mKdV lattice equation, the discretized nonlinear Schrödinger equation and the Toda lattice equation. By comparing the approximate solutions with the exact analytical solutions, we find the extend method for NDDEs is of good accuracy.     

The Discrete Relativistic Toda Molecule Equation and a Padé Approximation Algorithm

The relativistic Toda molecule equation (RTM) describes a one-parameter deformation of coefficients of the recurrence relation of a class of biorthogonal polynomials including the Szegö polynomials. In this paper, we present (i) explicit solutions of the discrete relativistic Toda molecule equation (d-RTM), (ii) a new Padé approximation algorithm for a given power series.     

Asymptotics for Supersonic Soliton Propagation in the Toda Lattice Equation

We study the problem of the adjustment of an initial condition to an exact supersonic soliton solution of the Toda latice equation. Also, we study the problem of soliton propagation in the Toda lattice with slowly varying mass impurities. In both cases we obtain the full numerical solution of the soliton evolution and we develop a modulation theory based on the averaged Lagrangian of the discrete Toda equation. Unlike previous problems with coherent subsonic solutions we need to modify the averaged Lagrangian to obtain the coupling between the supersonic soliton and the subsonic linear radiation. We show how this modified modulation theory explains qualitatively in simple terms the evolution of a supersonic soliton in the presence of impurities. The quantitative agreement between the modulation solution and the numerical result is good.     

2D Toda lattice equation with self-consistent sources: Casoratian type solutions,bilinear Bäcklund transformation and Lax pair

The two-dimensional Toda lattice equation with self-consistent sources is proposed based on its bilinear forms. Casoratian-type solutions and Bäcklund transformation (BT) for the bilinear forms are presented. Starting from the BT, a Lax pair is derived for the 2D Toda lattice with self-consistent sources.     

N=(1|1) Supersymmetric Dispersionless Toda Lattice Hierarchy

Generalizing the graded commutator in superalgebras, we propose a new bracket operation on the space of graded operators with an involution. We study properties of this operation and show that the Lax representation of the two-dimensional N=(1|1) supersymmetric Toda lattice hierarchy can be realized via the generalized bracket operation; this is important in constructing the semiclassical (continuum) limit of this hierarchy. We construct the continuum limit of the N=(1|1) Toda lattice hierarchy, the dispersionless N=(1|1) Toda hierarchy. In this limit, we obtain the Lax representation, with the generalized graded bracket becoming the corresponding Poisson bracket on the graded phase superspace. We find bosonic symmetries of the dispersionless N=(1|1) supersymmetric Toda equation.     

The Darboux transformation associated with two-parameter lattice soliton equation

A new discrete matrix spectral problem with two arbitrary constants is introduced. The corresponding two-parameter integrable lattice soliton equation is obtained through the discrete zero curvature representation, and the resulting integrable lattice equation reduce to the Toda lattice in rational form for a special choice of the parameters. A Darboux transformation (DT) for the lattice soliton equation is constructed. As an application, an explicit solution of the two-parameter lattice soliton equation is presented.     

Singly Periodic Solutions of the Allen-Cahn Equation and the Toda Lattice

The Allen-Cahn equation ? Δu = u ? u ³ in ?² has family of trivial singly periodic solutions that come from the one dimensional periodic solutions of the problem ?u″ =u ? u ³. In this paper we construct a non-trivial family of singly periodic solutions to the Allen-Cahn equation. Our construction relies on the connection between this equation and the infinite Toda lattice. We show that for each one-soliton solution to the infinite Toda lattice we can find a singly periodic solution to the Allen-Cahn equation, such that its level set is close to the scaled one-soliton. The solutions we construct are analogues of the family of Riemann minimal surfaces in ?³.     

Integrals of open two-dimensional lattices

We present an explicit formula for integrals of the open two-dimensional Toda lattice of type A_n. This formula is applicable for various reductions of this lattice. As an illustration, we find integrals of the G₂ Toda lattice. We also reveal a connection between the open An Toda and Shabat-Yamilov lattices.     

Webs, Lenard schemes, and the local geometry of bi-Hamiltonian Toda and Lax structures

We introduce a criterion that a given bi-Hamiltonian structure admits a local coordinate system where both brackets have constant coefficients. This criterion is applied to the bi-Hamiltonian open Toda lattice in a generic point, which is shown to be locally isomorphic to a Kronecker odd-dimensional pair of brackets with constant coefficients. This shows that the open Toda lattice cannot be locally represented as a product of two bi-Hamiltonian structures. Near, a generic point, the bi-Hamiltonian periodic Toda lattice is shown to be isomorphic to a product of two open Toda lattices (one of which is a (trivial) structure of dimension 1). While the above results might be obtained by more traditional methods, we use an approach based on general results on geometry of webs. This demonstrates the possibility of applying a geometric language to problems on bi-Hamiltonian integrable systems; such a possibility may be no less important than the particular results proved in this paper. Based on these geometric approaches, we conjecture that decompositions similar to the decomposition of the periodic Toda lattice exist in local geometry of the Volterra system, the complete Toda lattice, the multidimensional Euler top, and a regular bi-Hamiltonian Lie coalgebra. We also state general conjectures about the geometry of more general "homogeneous" finite-dimensional bi-Hamiltonian structures. The class of homogeneous structures is shown to coincide with the class of systems integrable by Lenard scheme. The bi-Hamiltonian structures which admit a non-degenerate Lax structure are shown to be locally isomorphic to the open Toda lattice.     

Noether and master symmetries for the Toda lattice

In this letter we examine the interrelation between Noether symmetries, master symmetries and recursion operators for the Toda lattice. The topics include invariants, higher Poisson brackets and the various relations they satisfy. For the case of two degrees of freedom we prove that the Toda lattice is super-integrable.     

Korteweg-de Vries and nonlinear equations related to the Toda lattice

The purpose of this work is to discuss the sense in which certain nonlinear equations related to the Toda lattice are discrete versions of the Korteweg-de Vries equation.     

Involutive solutions for toda and langmuir lattices through nonlinearization of lax pairs

Under the constraint determined by a relation (a _n ,b _n )^T={f(?)} _n between the reflectionless potentials and the eigenfunctions of the general discrete Schrödinger eigenvalue problem, the Lax pair of the Toda lattice is nonlinearized to be a finite-dimensional difference system and a nonlinear evolution equation, while the solution varietyN of the former is an invariant set of S-flows determined by the latter, and the constants of the motion for the algebraic system are presented.f maps the solution of the algebraic system into the solution of a certain stationary Toda equation. Similar results concerning the Langmuir lattice are given, and a relation between the two difference systems, which are the spatial parts of the nonlinearized Lax pairs of the Toda lattice and Langmuir lattice, is discussed.     

Toda-type differential equations for the recurrence coefficients of orthogonal polynomials and Freud transformation

Transformations of the measure of orthogonality for orthogonal polynomials, namely Freud transformations, are considered. Jacobi matrix of the recurrence coefficients of orthogonal polynomials possesses an isospectral deformation under these transformations. Dynamics of the coefficients are described by generalized Toda equations. The classical Toda lattice equations are the simplest special case of dynamics of the coefficients under the Freud transformation of the measure of orthogonality. Also dynamics of Hankel determinants, its minors and other notions corresponding to the orthogonal polynomials are studied.     

Two-periodic waves and asymptotic property for generalized 2D Toda lattice equation

In this paper, two-periodic wave solutions are constructed for the (2 + 1)-dimensional generalized Toda lattice equation by using Hirota bilinear method and Riemann theta function. At the same time, we analyze in details asymptotic properties of the two-periodic wave solutions and give their asymptotic relations between the periodic wave solutions and the soliton solutions.     

Coupled nonlinear waves in two-dimensional lattice

For one-dimensional nonlinear lattices, such as Toda lattice, it has been extensively studied. By considering the nonlinear effects of two-dimensional lattice, we set up the equation of motion for each particles (atoms, molecules or ions). For small amplitude and long wavelength nonlinear waves in this system, both the linear dispersion relation and the coupled Korteweg de Vries (KdV) equation are obtained. The simple soliton solution is obtained. If the nonlinear lattice is symmetric in the x and y directions, It is noted that there are two kinds of solitons. one is that propagates in either x or y directions, (1, 0) or (0, 1), the other is that propagates in the direction of (1, 1). It is in agreements with that of one-dimensional lattice. The different properties are investigated for different nonlinear interacting potentials, such as Toda potential, Morse potential and LJ potential.     

Singular solutions in Casoratian form for two differential-difference equations

Negatons, positons, rational solutions and mixed solutions in Casoratian form for the Toda lattice and the differential-difference KdV equation are obtained. Some characteristics of the obtained singular solutions are investigated through density graphics.     

Complex high order Toda and Volterra lattices1

Given a solution of a high order Toda lattice we construct a one parameter family of new solutions. In our method, we use a set of Bäcklund transformations such that each new generalized Toda solution is related to a generalized Volterra solution.