Non-Lie Symmetry Groups of （2＋1）-Dimensional Nonlinear Systems

A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems. Applying the modified direct method to the well-known （2＋1）-dimensional asymmetric Nizhnik-Novikov-Vesselov equation and Nizhnik Novikov-Vesselov equation, both the Lie point symmetry groups and the non-Lie symmetry groups are obtained. The Lie symmetry groups obtained via traditional Lie approaches are only speciai cases. Furthermore, the expressions of the exact finite transformations of the Lie groups are much simpler than those obtained via the standard approaches.     

Symmetry Reductions and Group-Invariant Solutions of （2＋1）-Dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

With the aid of symbolic computation, we present the symmetry transformations of the （2＋ 1）-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada equation with Lou＇s direct method that is based on Lax pairs. Moreover, with the symmetry transformations we obtain the Lie point symmetries of the CDGKS equation, and reduce the equation with the obtained symmetries. As a result, three independent reductions are presented and some group-invariant solutions of the equation are given.     

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 Reductions of a Lax Pair for （2＋1）-Dimensional Differential Sawada-Kotera Equation

Based on the symbolic computational system Maple, the similarity reductions of a Lax pair for the （2＋1 ）-dimensional differential Sawada Kotera （SK） equation by the classical Lie point group method, are presented. We obtain several interesting reductions. Comparing the reduced Lax pair＇s compatibility with the reduced SK equation under the same symmetry group, we find that the reduced Lax pairs do not always lead to the reduced SK equation. In general, the reduced equations are the subsets of the compatibility conditions of the reduced Lax pair.     

A Maple Package to Compute Lie Symmetry Groups and Symmetry Reductions of （1＋1）-Dimensional Nonlinear Systems

Armed with the computer algebra system Maple, using a direct algebraic substitution method, we obtain Lie point symmetries, Lie symmetry groups and the corresponding symmetry reductions of one component nonlinear integrable and nonintegrable equations only by clicking the ‘Enter＇ key. Abundant （1＋1）-dimensional nonlinear mathematical physical systems are analysed effectively by using a Maple package LieSYMGRP proposed by us.     

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.     

Symmetry Groups and Exact Solutions of New （4＋1）-Dimensional Fokas Equation

In this paper, we investigate symmetries of the new （4＋1）-dimensional Fokas equation, including point symmetries and the potential symmetries. We firstly employ the algorithmic procedure of computing the point symmetries. And then we transform the Fokas equation into a potential system and gain the potential symmetries of Fokas equation. Finally, we use the obtained point symmetries wave solutions and other solutions of the Fokas equation. and some constructive methods to get some doubly periodic In particular, some solitary wave solutions are also given.     

Similarity Reductions of （2＋1）-Dimensional Multi-component Broer-Kaup System

Painleve property of the （2＋1）-dimensional multi-component Broer-Kaup （BK） system is considered by using the standard Weiss Kruskal approaches. Applying the Clarkson and Kruskal （CK） direct method to the （2＋1）- dimensional multi-component BK system, some types of similarity reductions are obtained. By solving the reductions, one can get the solutions of the （2＋1）-dimensional multi-component BK system.     

Symmetry and Similarity Solutions of a （2＋1）-Dimensional Generalized Broer-Kaup System

We study the symmetries of a （2＋1）-dimensional generalized Broer-Kaup system by means of the classical Lie group theory. The corresponding group algebra is constructed. Based on the symmetries, severaJ types of similarity solutions are obtained.     

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.     

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.     

Lie Symmetry Groups of （2＋1）-Dimensional BKP Equation and Its New Solutions

A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems. Applying the modified direct method to the well-known （2＋1）-dimensional B KP equation we get its symmetry. Furthermore, the exact solutions of （2＋1）-dimensional BKP equation are obtained through symmetry analysis.     

Symmetries and Similarity Reductions of a New （2＋1）-Dimensional Shallow Water Wave System

The symmetries of a （2＋1）-dimensional shallow water wave system, which is newly constructed through applying variation principle of analytic mechanics, are researched in this paper. The Lie symmetries and the corresponding reductions are obtained by means of classical Lie group approach. The （1＋1） dimensional displacement shallow water wave equation can be derived from the reductions when special symmetry parameters are chosen.     

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.     

Symmetries and Algebras of a （2＋1）-Dimensional MKdV-Type System

In this paper, a （2＋1）-dimensional MKdV-type system is considered. By applying the formal series symmetry approach, a set of infinitely many generalized symmetries is obtained. These symmetries constitute a closed infinite-dimensional Lie algebra which is a generalization of w∞ type algebra. Thus the complete integrability of this system is confirmed.     

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 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.     

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.     

Symmetry Analysis and Exact Solutions of （2＋1）-Dimensional Sawada-Kotera Equation

Based on the symbolic computation system Maple, the infinite-dimensional symmetry group of the （2＋1）- dimensional Sawada-Kotera equation is found by the classical Lie group method and the characterization of the group properties is given. The symmetry groups are used to perform the symmetry reduction. Moreover, with Lou＇s direct method that is based on Lax pairs, we obtain the symmetry transformations of the Sawada-Kotera and Konopelchenko Dubrovsky equations, respectively.     

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.