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.     

Symmetry Groups and New Exact Solutions to （2＋1）-Dimensional Variable Coefficient Canonical Generalized KP Equation

In this paper, the modified CK＇s direct method to find symmetry groups of nonlinear partial differential equation is extended to （2＋1）-dimensional variable coeffficient canonical generalized KP （VCCGKP） equation. As a result, symmetry groups, Lie point symmetry group and Lie symmetry for the VCCGKP equation are obtained. In fact, the Lie point symmetry group coincides with that obtained by the standard Lie group approach. Applying the given Lie symmetry, we obtain five types of similarity reductions and a lot of new exact solutions, including hyperbolic function solutions, triangular periodic solutions, Jacobi elliptic function solutions and rational solutions, for the VCCGKP equation.     

Symmetry Reduction, Exact Solutions, and Conservation Laws of （2＋1）-Dimensional Burgers Korteweg-de Vries Equation

Using the classical Lie method of infinitesimals, we first obtain the symmetry of the （2＋1）-dimensional Burgers-Korteweg-de-Vries （3D-BKdV） equation. Then we reduce the 3D-BKdV equation using the symmetry and give some exact solutions of the 3D-BKdV equation. When using the direct method, we restrict a condition and get a relationship between the new solutions and the old ones. Given a solution of the 3D-BKdV equation, we can get a new one from the relationship. The relationship between the symmetry obtained by using the classical Lie method and that obtained by using the direct method is also mentioned. At last, we give the conservation laws of the 3D-BKdV equation.     

Travelling Wave Solutions for Konopelchenko-Dubrovsky Equation Using an Extended sinh-Gordon Equation Expansion Method

The sinh-Gordon equation expansion method is further extended by generMizing the sinh-Gordon equation and constructing new ansatz solution of the considered equation. As its application, the （2＋1）-dimensional Konopelchenko-Dubrovsky equation is investigated and abundant exact travelling wave solutions are explicitly obtained including solitary wave solutions, trigonometric function solutions and Jacobi elliptic doubly periodic function solutions, some of which are new exact solutions that we have never seen before within our knowledge. The method can be applied to other nonlinear evolution equations in mathematical physics.     

Symmetry Reduction and Exact Solutions of the （3＋1）-Dimensional Breaking Soliton Equation

By means of the generalized direct method, a relationship is constructed between the new solutions and the old ones of the （3＋1）-dimensional breaking soliton equation. Based on the relationship, a new solution is obtained by using a given solution of the equation. The symmetry is also obtained for the （3＋1）-dimensional breaking soliton equation. By using the equivalent vector of the symmetry, we construct a seven-dimensional symmetry algebra and get the optimal system of group-invariant solutions. To every case of the optimal system, the （3＋1）-dimensional breaking soliton equation is reduced and some solutions to the reduced equations are obtained. Furthermore, some new explicit solutions are found for the （3＋ 1）-dimensional breaking soliton equation.     

Exact Solutions of （2＋1）-Dimensional Boiti-Leon-Pempinelle Equation with （G′/G）-Expansion Method

In this paper, we construct exact solutions for the （2＋1）-dimensional Boiti-Leon-Pempinelle equation by using the （G′/G）-expansion method, and with the help of Maple. As a result, non-travelling wave solutions with three arbitrary functions are obtained including hyperbolic function solutions, trigonometric function solutions, and rational solutions. This method can be applied to other higher-dimensional nonlinear partial differential equations.     

Two Classes of New Exact Solutions to （2＋1）-Dimensional Breaking Soliton Equation

The singular manifold method is used to obtain two general solutions to a （2＋1）-dimensional breaking soliton equation, each of which contains two arbitrary functions. Then the new periodic wave solutions in terms of the Jacobi elliptic functions are generated from the general solutions. The long wave limit yields the new types of dromion and solitary structures.     

Finite Symmetry Transformation Groups and Some Exact Solutions to （2＋1）-Dimensional Cubic Nonlinear SchrSdinger Equantion

Making use of the direct method proposed by Lou et al. and symbolic computation, finite symmetry transformation groups for a （2＋ l）-dimensional cubic nonlinear Schrodinger （NLS） equation and its corresponding cylindrical NLS equations are presented. Nine related linear independent infinitesimal generators can be obtained from the finite symmetry transformation groups by restricting the arbitrary constants in infinitesimal forms. Some exact solutions are derived from a simple travelling wave solution.     

New Exact Periodic Solitary-Wave Solutions for New （2＋1）-Dimensional KdV Equation

The extended homoclinic test function method is a kind of classic, efficient and well-developed method to solve nonlinear evolution equations. In this paper, with the help of this approach, we obtain new exact solutions （including kinky periodic solitary-wave solutions, periodic soliton solutions, and cross kink-wave solutions） for the new （2＋1）-dimensional KdV equation. These results enrich the variety of the dynamics of higher-dimensionai nonlinear wave field.     

Symmetry and Exact Solutions of （2＋1）-Dimensional Generalized Sasa-Satsuma Equation via a Modified Direct Method

In this paper, first, we employ classic Lie symmetry groups approach to obtain the Lie symmetry groups, of the well-known （2＋1）-dimensional Generalized Sasa-Satsuma （GSS） equation. Second, based on a modified direct method proposed by Lou [J.Phys.A： Math. Gen. 38 （2005） L129], more general symmetry groups are obtained and the relationship between the new solution and known solution is set up. At the same time, the Lie symmetry groups obtained are only special cases of the more general symmetry groups. At last, some exact solutions of GSS equations are constructed by the relationship obtained in the paper between the new solution and known solution.     

Symmetry Groups and New Exact Solutions of （2＋1）-Dimensional Dispersive Long-Wave Equations

Using the modified CK＇s direct method, we derive a symmetry group theorem of （2＋1）-dimensional dispersive long-wave equations. Based upon the theorem, Lie point symmetry groups and new exact solutions of （2＋1）- dimensional dispersive long-wave equations are obtained.     

Kac-Moody-Virasoro Symmetry Algebra of （2＋1）-Dimensional Dispersive Long-Wave Equation with Arbitrary Order Invariant

By Lie symmetry method, the Lie point symmetries and its Kac Moody-Virasoro （KMV） symmetry algebra of （2＋1）-dimensional dispersive long-wave equation （DLWE） are obtained, and the finite transformation of DLWE is given by symmetry group direct method, which can recover Lie point symmetries. Then KMV symmetry algebra of DLWE with arbitrary order invariant is also obtained. On basis of this algebra the group invariant solutions and similarity reductions are also derived.     

A Series of Exact Solutions for a New （2＋1）-Dimensional Calogero KdV Equation

An algebraic method is proposed to solve a new （2＋1）-dimensional Calogero KdV equation and explicitly construct a series of exact solutions including rational solutions, triangular solutions, exponential solution, line soliton solutions, and doubly periodic wave solutions.     

New exact periodic solutions to （2＋1）-dimensional dispersive long wave equations

In this paper, we make use of the auxiliary equation and the expanded mapping methods to find the new exact periodic solutions for （2＋1）-dimensional dispersive long wave equations in mathematical physics, which are expressed by Jacobi elliptic functions, and obtain some new solitary wave solutions （m → 1）. This method can also be used to explore new periodic wave solutions for other nonlinear evolution equations.     

A Simple Method to Generate Lie Point Symmetry Groups of the （3＋1）-Dimensional Jimbo-Miwa Equation

Taking the (3 1)-dimensional Jimbo-Miwa equation as a simple example, we develop a modified direct method to find symmetry groups and symmetry algebras. Some exact solutions of the model are given by the simple method.     

Study of （2＋1）-Dimensional Higher-Order Broer-Kaup System

Painleve property and infinite symmetries of the （2＋1）-dimensional higher-order Broer-Kaup （HBK） system are studied in this paper. Using the modified direct method, we derive the theorem of general symmetry gro.ups to （2＋1）-dimensional HBK system. Based on our theorem, some new forms of solutions are obtained. We also find infinite number of conservation laws of the （2＋1）-dimensional HBK system.     

A new variable coefficient algebraic method and non-travelling wave solutions of nonlinear equations

In this paper, a new auxiliary equation method is presented of constructing more new non-travelling wave solutions of nonlinear differential equations in mathematical physics, which is direct and more powerful than projective Riccati equation method. In order to illustrate the validity and the advantages of the method, （2＋1）-dimensional asymmetric Nizhnik-Novikov-Vesselov equation is employed and many new double periodic non-travelling wave solutions are obtained. This algorithm can also be applied to other nonlinear differential equations.     

A novel （G＇/G）-expansion method and its application to the Boussinesq equation

In this article, a novel （G＇/G）-expansion method is proposed to search for the traveling wave solutions of nonlinear evolution equations. We construct abundant traveling wave solutions involving parameters to the Boussinesq equation by means of the suggested method. The performance of the method is reliable and useful, and gives more general exact solutions than the existing methods. The new （G＇/G）-expansion method provides not only more general forms of solutions but also cuspon, peakon, soliton, and periodic waves.     

Symmetry, Reductions and New Exact Solutions of ANNV Equation Through Lax Pair

In this paper, the direct symmetry method is extended to the Lax pair of the ANNV equation. As a result, symmetries of the Lax pair and the ANNV equation are obtained at the same time. Applying the obtained symmetry, the （2＋1）-dimensional Lax pair is reduced to （1＋1）-dimensional Lax pair, whose compatibility yields the reduction of the ANNV equation. Based on the obtained reductions of the ANNV equation, a lot of new exact solutions for the ANNV equation are found. This shows that for an integrable system, both the symmetry and the reductions can be obtained through its corresponding Lax pair.     

Symmetry Analysis of Two Types of （2＋1）-Dimensional Nonlinear Klein-Gorden Equation

By means of the classical symmetry method, we investigate two types of the （2＋1）-dimensional nonlinear Klein-Gorden equation. For the wave equation, we give out its symmetry group analysis in detail. For the second type of the （2＋1）-dimensional nonlinear Klein-Gorden equation, an optimal system of its one-dimensional subalgebras is constructed and some corresponding two-dimensional symmetry reductions are obtained.