Hierarchy of Combined TL-RTL Equations and an Associated (2+1)-Dimensional Lattice Equation

A new (2+1)-dimensional lattice equation is presented based upon the first two members in the hierarchy of the combined Toda lattice and relativistic Toda lattice (TL-RTL) equations in (1+1) dimensions. A Darboux transformation for the hierarchy of the combined TL-RTL equations is constructed. Solutions of the first two members in the hierarchy of the combined TL-RTL equations, as well as the new (2+1)-dimensional lattice equation are explicitly obtained by the Darboux transformation.     

（2＋2）-Dimensional Discrete Soliton Equations and Integrable Coupling System

In this paper, we extend a （2＋2）-dimensional continuous zero curvature equation to （2＋2）-dimensional discrete zero curvature equation, then a new （2＋2）-dimensional cubic Volterra lattice hierarchy is obtained. Fhrthermore, the integrable coupling systems of the （2＋2）-dimensional cubic Volterra lattice hierarchy and the generalized Toda lattice soliton equations are presented by using a Lie algebraic system sl（4）.     

Constructing New Discrete Integrable Coupling System for Soliton Equation by Kronecker Product

It is shown that the Kronecker product can be applied to constructing new discrete integrable coupling system of soliton equation hierarchy in this paper. A direct application to the fractional cubic Volterra lattice spectral problem leads to a novel integrable coupling system of soliton equation hierarchy. It is also indicated that the study of discrete integrable couplings by using the Kronecker product is an efficient and straightforward method. This method can be used generally.     

利用耦合的Riccati方程组构造微分-差分方程精确解

通过引入耦合的Riccati方程组得到一个构造非线性微分-差分方程精确解的代数方法.作为实例,将该方法应用到了一般格子方程,相对论的Toda格子方程和(2+1)维Toda格子方程.借助符号计算软件Mathematica,获得了这些方程的扭结型孤波解和复数解.该方法也适合求解其他非线性微分-差分方程的精确解. 关键词： 耦合Riccati方程组 格子方程 相对论的Toda格子方程 (2+1)维Toda格子方程     

Hierarchy of Combined TL-RTL Equations and an Associated (2+1)-Dimensional Lattice Equation

A new (2 1)-dimensional lattice equation is presented based upon the first two members in the hierarchy of the combined Toda lattice and relativistic Toda lattice (TL-RTL) equations in (1 1) dimensions. A Darboux transformation for the hierarchy of the combined TL-RTL equations is constructed. Solutions of the first two members in the hierarchy of the combined TL-RTL equations, as well as the new (2 1)-dimensional lattice equation are explicitly obtained by the Darboux transformation.     

Casoratian Solutions to Toda Lattice Via Its Backlund Transformation

Generalized Casoratian condition and Casoratian solutions of the Toda lattice are given in terms of its bilinear Bgcklund transformation. By choosing suitable Casoratian entries and parameter in the bilinear Bgcklund transformation, we can give transformations among many kinds of solutions.     

On （2＋1）-Dimensional Non-isospectral Toda Lattice Hierarchy and Integrable Coupling System

By considering （2＋1）-dimensional non-isospectral discrete zero curvature equation, the （2＋1）-dimensional non-isospectral Toda lattice hierarchy is constructed in this article. It follows that some reductions of the （2＋1）- dimensional Toda lattice hierarchy are given. Finally, the （2＋1）-dimensional integrable coupling system of the Toda lattice hierarchy is obtained through enlarging spectral problem.     

New integrable lattice hierarchies

In this Letter we give a new integrable four-field lattice hierarchy, associated to a new discrete spectral problem. We obtain our hierarchy as the compatibility condition of this spectral problem and an associated equation, constructed herein, for the time-evolution of eigenfunctions. We consider reductions of our hierarchy, which also of course admit discrete zero curvature representations, in detail. We find that our hierarchy includes many well-known integrable hierarchies as special cases, including the Toda lattice hierarchy, the modified Toda lattice hierarchy, the relativistic Toda lattice hierarchy, and the Volterra lattice hierarchy. We also obtain here a new integrable two-field lattice hierarchy, to which we give the name of Suris lattice hierarchy, since the first equation of this hierarchy has previously been given by Suris. The Hamiltonian structure of the Suris lattice hierarchy is obtained by means of a trace identity formula.     

Two new discrete integrable systems

In this paper,we focus on the construction of new(1+1)-dimensional discrete integrable systems according to a subalgebra of loop algebra A 1.By designing two new(1+1)-dimensional discrete spectral problems,two new discrete integrable systems are obtained,namely,a 2-field lattice hierarchy and a 3-field lattice hierarchy.When deriving the two new discrete integrable systems,we find the generalized relativistic Toda lattice hierarchy and the generalized modified Toda lattice hierarchy.Moreover,we also obtain the Hamiltonian structures of the two lattice hierarchies by means of the discrete trace identity.     

Application of Exp-Function Method to Discrete Nonlinear SchrSdinger Lattice Equation with Symbolic Computation

In this paper, we present an extended Exp-function method to differential-difference equation（s）. With the help of symbolic computation, we solve discrete nonlinear Schrodinger lattice as an example, and obtain a series of general solutions in forms of Exp-function.     

Symmetry Reductions and Exact Solutions of Blaszak-Marciniak Four-Field Lattice Equation

Applying the Lie group method to the differential-difference equation, the Lie point symmetry of Blaszak- Marciniak four-field Lattice equation is obtained. Using the obtained symmetry, the similarity reduction equations of Blaszak-Marciniak four-field Lattice equation are derived. Solving the reduction, we get the solution of Blaszak-Marciniak four-field Lattice equation which not only recovers one of the solutions obtained by Ma and Hu [J. Math. Phys. 40 （1999） 6071] but also has the singularity when we choose the arbitrary constants accurately.     

非线性离散微分方程的双曲函数法求解

本文推广了双曲函数方法用于求解非线性离散系统。求解离散的(2+1)维Toda系统和离散的mKdV系统，成功地得到了离散钟型孤立子、离散冲击波型孤立子及一些新的精确行波解。     

Darboux–Bäcklund transformation and explicit solutions to a hybrid lattice of the relativistic Toda lattice and the modified Toda lattice

The relativistic Toda lattice and the modified Toda lattice are two important discrete integrable equations. In this paper, we investigate a hybrid lattice equation of the two lattice equations. Darboux–Bäcklund transformation and explicit solutions to the hybrid lattice equation are constructed.     

A New Generalized System on the Two-Dimensional Toda Lattice and Its Reduction

In this paper, we apply the source generation procedure to the coupled 2D Toda lattice equation (also called Pfaffianized 2D Toda lattice), then we get a more generalized system which is the coupled 2D Toda lattice with self-consistent sources (p-2D TodaESCS), and a pfaffian type solution of the new system is given. Consequently, by using the reduction of the pfaffian solution to the determinant form, this new system can not only be reduced to the 2D TodaESCS, but be reduced to the coupled 2D Toda lattice equation. This result indicates that the p-2D TodaESCS is also a pfaffian version of the 2D TodaESCS, which implies the commutativity between the source generation procedure and Pfaffianization is valid to the semi-discrete soliton equation.     

Solitary Wave and Periodic Wave Solutions for the Relativistic Toda Lattices

In this work, an adaptation of the tanh/tan-method that is discussed usually in the nonlinear partial differential equations is presented to solve nonlinear polynomial differential-difference equations. As a concrete example, several solitary wave and periodic wave solutions for the chain which is related to the relativistic Toda lattice are derived. Some systems of the differential-difference equations that can be solved using our approach are listed and a discussion is given in conclusion.     

Breather Lattice Solutions to Negative mKdV Equation

In this paper, dependent and independent variable transformations are introduced to solve the negative mKdV equation systematically by using the knowledge of elliptic equation and Jacobian elliptic functions. It is shown that different kinds of solutions can be obtained to the negative mKdV equation, including breather lattice solution and periodic wave solution.     

What is the Relativistic Volterra Lattice?

We develop a systematic procedure of finding integrable 6ldquo;relativistic” (regular one-parameter) deformations for integrable lattice systems. Our procedure is based on the integrable time discretizations and consists of three steps. First, for a given system one finds a local discretization living in the same hierarchy. Second, one considers this discretization as a particular Cauchy problem for a certain 2-dimensional lattice equation, and then looks for another meaningful Cauchy problem, which can be, in turn, interpreted as a new discrete time system. Third, one has to identify integrable hierarchies to which these new discrete time systems belong. These novel hierarchies are called then “relativistic”, the small time step $h$ playing the role of inverse speed of light. We apply this procedure to the Toda lattice (and recover the well-known relativistic Toda lattice), as well as to the Volterra lattice and a certain Bogoyavlensky lattice, for which the “relativistic” deformations were not known previously. Received: 1 April 1998 / Accepted: 21 July 1998     

New homotopy analysis transform method for solving the discontinued problems arising in nanotechnology

We present a new reliable analytical study for solving the discontinued problems arising in nanotechnology.Such problems are presented as nonlinear differential–difference equations.The proposed method is based on the Laplace transform with the homotopy analysis method(HAM).This method is a powerful tool for solving a large amount of problems.This technique provides a series of functions which may converge to the exact solution of the problem.A good agreement between the obtained solution and some well-known results is obtained.     

Hamiltonian Forms for a Hierarchy of Discrete Integrable Coupling Systems

A semi-direct sum of two Lie algebras of four-by-four matrices is presented, and a discrete four-by-four matrix spectral problem is introduced. A hierarchy of discrete integrable coupling systems is derived. The obtained integrable coupling systems are all written in their Hamiltonian forms by the discrete variational identity. Finally, we prove that the lattice equations in the obtained integrable coupling systems are all Liouville integrable discrete Hamiltonian systems.     

Explicit Solutions for a （2＋1）-Dimensional Toda Lattice with Two Discrete Variables

An integrable （2＋1）-dimensional Toda lattice with two discrete variables is investigated again, which is produced from a compatible condition of the Lax triad. The Darboux transformation for its spectral problems is found. As an application, explicit solutions of the （2＋1）-dimensional Toda equation with two discrete variables are obtained.