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.     

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

A new （29-1）-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 （19991） 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 （29-1）-dimensional lattice equation are explicitly obtained by the Darboux transformation.     

Integrable Coupling of （2＋1）-Dimensional Multi-component DLW Integrable Hierarchy and Its Hamiltonian Structure

A new multi-component Lie algebra is constructed, and a type of new loop algebra is presented. A （2＋1）-dimensional multi-component DLW integrable hierarchy is obtained by using a （2＋1）-dimensional zero curvature equation. Furthermore, the loop algebra is expanded into a larger one and a type of integrable coupling system and its corresponding Hamiltonian structure are worked out.     

The multicomponent （2＋1）-dimensional Glachette-Johnson （GJ） equation hierarchy and its super-integrable coupling system

This paper presents a set of multicomponent matrix Lie algebra, which is used to construct a new loop algebra A^-M. By using the Tu scheme, a Liouville integrable multicomponent equation hierarchy is generated, which possesses the Hamiltonian structure. As its reduction cases, the multicomponent （2＋1）-dimensional Glachette-Johnson （G J） hierarchy is given. Finally, the super-integrable coupling system of multicomponent （2＋1）-dimensional GJ hierarchy is established through enlarging the spectral problem.     

Hamiltonian System of New Nonlinear Lattice Equations

A 3-dimensional Lie algebra sμ（3） is obtained with the help of the known Lie algebra. Based on the sμ（3）, a new discrete 3 × 3 matrix spectral problem with three potentials is constructed. In virtue of discrete zero curvature equations, a new matrix Lax representation for the hierarchy of the discrete lattice soliton equations is acquired. It is shown that the hierarchy possesses a Hamiltonian operator and a hereditary recursion operator, which implies that there exist infinitely many common commuting symmetries and infinitely many common commuting conserved functionals.     

Rational and Periodic Solutions for a （2＋1）-Dimensional Breaking Soliton Equation Associated with ZS-AKNS Hierarchy

The double Wronskian solutions whose entries satisfy matrix equation for a （2＋1）-dimensional breaking soliton equation （（2＋ 1）DBSE） associated with the ZS-AKNS hierarchy are derived through the Wronskian technique. Rational and periodic solutions for （2＋1）DBSE are obtained by taking special eases in general double Wronskian solutions.     

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.     

Noncommutative AKNS Equation Hierarchy and Its Integrable Couplings with Kronecker Product

We present a noncommutative version of the Ablowitz-Kaup-Newell-Segur （AKNS） equation hierarchy, which possesses the zero curvature representation. Furthermore, we derive the noncommutative AKNS equation from the noncommutative （anti-）self-dual Yang-Mills equation by reduction, which is an evidence for the noncommutative Ward conjecture. Finally, the integrable coupling system of the noncommutative AKNS equation hierarchy is constructed by using the Kronecker product.     

Algebraic-Geometric Solution to （2＋1）-Dimensional Sawada-Kotera Equation

Special solution of the （2＋1）-dimensional Sawada Kotera equation is decomposed into three （0＋1）- dimensional Bargmann flows. They are straightened out on the Jacobi variety of the associated hyperelliptic curve. Explicit algebraic-geometric solution is obtained on the basis of a deeper understanding of the KdV hierarchy.     

Generation of Nonlinear Evolution Equations by Reductions of the Self-Dual Yang–Mills Equations

With the help of some reductions of the self-dual Yang Mills （briefly written as sdYM） equations, we introduce a Lax pair whose compatibility condition leads to a set of （2 ＋ 1）-dimensional equations. Its first reduction gives rise to a generalized variable-coefficient Burgers equation with a forced term. Furthermore, the Burgers equation again reduces to a forced Burgers equation with constant coefficients, the standard Burgers equation, the heat equation, the Fisher equation, and the Huxley equation, respectively. The second reduction generates a few new （2 ＋ 1）-dimensional nonlinear integrable systems, in particular, obtains a kind of （2 ＋ 1）-dimensional integrable couplings of a new （2 ＋ 1）- dimensional integrable nonlinear equation.     

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.     

Exact Solutions of （2＋1）-Dimensional HNLS Equation

In this paper, we use the classical Lie group symmetry method to get the Lie point symmetries of the （2＋1）-dimensional hyperbolic nonlinear Schr6dinger （HNLS） equation and reduce the （2＋1）-dimensional HNLS equation to some （1 ＋ 1 ）-dimensional partial differential systems. Finally, many exact travelling solutions of the （2＋1）-dimensional HNLS equation are obtained by the classical Lie symmetry reduced method.     

Algebro-Geometric Solution to Two New （2＋1）-Dimensional Modified Kadomtsev-Petviashvili Equations

Two new （2＋1）-dimensional modified Kadomtsev-Petviashvili （mKP） equations are presented, which are related to a hierarchy of （1＋1）-dimensional soliton equations. Through the nonlinearization of Lax pair and the Riemann-Jacobi inversion technique, the algebro-geometric solutions of both the mKP equations are obtained.     

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.     

N-Soliton Solutions of （2＋1）-Dimensional Non-isospectral AKNS System

The bilinear form of the （2＋1）-dimensional non-isospectral AKNS system is derived. Its N-soliton solutions are obtained by using the Hirota method. As a reduction, a （2＋1）-dimensional non-isospectral Schrodinger equation and its N-soliton solutions are constructed.     

New Positive and Negative Hierarchies of Integrable Differential-Difference Equations and Conservation Laws

Two hierarchies of nonlinear integrable positive and negative lattice equations are derived from a discrete spectrak problem. The two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. Moreover, the integrable lattice models in the positive hierarchy are of polynomial type, and the integrable lattice models in the negative hierarchy are of rational type. Further, we construct infinite conservation laws about the positive hierarchy.     

On High-Frequency Soliton Solutions to a （2＋1）-Dimensional Nonlinear Partial Differential Evolution Equation

A （2＋1）-dimensional nonlinear partial differential evolution （NLPDE） equation is presented as a model equation for relaxing high-rate processes in active barothropic media. With the aid of symbolic computation and Hirota＇s method, some typical solitary wave solutions to this （2＋1）-dimensional NLPDE equation are unearthed. As a result, depending on the dissipative parameter, single and multivalued solutions are depicted.     

Exact Discrete Soliton Solutions and Periodic Solutions to （2＋1）-Dimensional Toda Lattice with a New Algebraic Method

In this paper, with the aid of symbolic computation, we present a uniform method for constructing soliton solutions and periodic solutions to （2＋1）-dimensional Toda lattice equation.     

New Multi-soliton Solutions for the （2＋1）-Dimensional Kadomtsev-Petviashvili Equation

In this paper, an explicit N-fold Darboux transformation with multi-parameters for both a （1＋1）- dimensional Broer-Kaup （BK） equation and a （1＋1）-dimensional high-order Broer-Kaup equation is constructed with the help of a gauge transformation of their spectral problems. By using the Darboux transformation and new basic solutions of the spectral problems, 2N-soliton solutions of the BK equation, the high-order BK equation, and the Kadomtsev-Petviashvili （KP） equation are obtained.     

Forced(2+1)-dimensional discrete three-wave equation

We generalize the ■-dressing method to investigate a(2+1)-dimensional lattice,which can be regarded as a forced(2+1)-dimensional discrete three-wave equation.The soliton solutions to the(2+1)-dimensional lattice are given through constructing different symmetry conditions.The asymptotic analysis of one-soliton solution is discussed.For the soliton solution,the forces are zero.