Exact Solutions to （2＋1）-Dimensional Kaup-Kupershmidt Equation

In this paper, by using the symmetry method, the relationships between new explicit solutions and old ones of the （2＋1）-dimensional Kaup-Kupershmidt （KK） equation are presented. We successfully obtain more general exact travelling wave solutions for （2＋ 1）-dimensional KK equation by the symmetry method and the （G1/G）-expansion method. Consequently, we find some new solutions of （2＋1）-dimensional KK equation, including similarity solutions, solitary wave solutions, and periodic solutions.     

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.     

Double Elliptic Equation Expansion Approach and Novel Solutions of （2＋1）-Dimensional Break Soliton Equation

In this paper, by means of double elliptic equation expansion approach, the novel double nonlinear wave solutions of the （2＋1）-dimensional break soliton equation are obtained. These double nonlinear wave solutions contain the double Jacobi elliptic function-like solutions, the double solitary wave-like solutions, and so on. The method is also powerful to some other nonlinear wave equations in （2＋1） dimensions.     

Periodic Wave Solution to the （3＋1）-Dimensional Boussinesq Equation

One- and two-periodic wave solutions for （3＋l）-dimensional Boussinesq equation are presented by means of Hirota＇s bilinear method and the Riemann theta function. The soliton solution can be obtained from the periodic wave solution in an appropriate limiting procedure.     

Analytical Solitary Wave Solutions to a （3＋1）-Dimensional Gross-Pitaevskii Equation with Variable Coefficients

A （3＋1）-dimensional Gross-Pitaevskii （GP） equation with time variable coefficients is considered, and is transformed into a standard nonlinear Schrodinger （NLS） equation. Exact solutions of the （3＋1）D GP equation are constructed via those of the NLS equation. By applying specific time-modulated nonlinearities, dispersions, and potentials, the dynamics of the solutions can be controlled. Solitary and periodic wave solutions with snaking and breathing behavior are reported.     

A Generalized Extended F-Expansion Method and Its Application in （2＋1）-Dimensional Dispersive Long Wave Equation

A new generalized extended F-expansion method is presented for finding periodic wave solutions of nonlinear evolution equations in mathematical physics. As an application of this method, we study the （2＋1）-dimensional dispersive long wave equation. With the aid of computerized symbolic computation, a number of doubly periodic wave solutions expressed by various Jacobi elliptic functions are obtained. In the limit cases, the solitary wave solutions are derived as well.     

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.     

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.     

New Doubly Periodic Waves of the （2＋1）-Dimensional Double Sine-Gordon Equation

New exact solutions of the （2 ＋1）-dimensional double sine-Gordon equation are studied by introducing the modified mapping relations between the cubic nonlinear Klein-Gordon system and double sine-Gordon equation. Two arbitrary functions are included into the Jacobi elliptic function solutions. New doubly periodic wave solutions are obtained and displayed graphically by proper selections of the arbitrary functions.     

New Periodic Solution to Jacobi Elliptic Functions of a （2d-1）-Dimensional BKP Equation and a Generalized Klein-Gordon Equation

With the help of the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method, the first elliptic function equation is used to obtain the Jacobi doubly periodic wave solutions of the （2＋1）-dimensional B-type Kadomtsev-Petviashvili （BKP） equation and the generalized Klein-Gordon equation. The method is also valid for other （1＋1）-dimensional and higher dimensional systems.     

New Types of Travelling Wave Solutions From （2＋l）-Dimensional Davey-Stewartson Equation

In this paper, based on new auxiliary nonlinear ordinary differential equation with a sixtb-aegree nonnneal term, we study the （2＋l ）-dimensional Davey-Stewartson equation and new types of travelling wave solutions are obtained, which include new bell and kink profile solitary wave solutions, triangular periodic wave solutions, and singular solutions. The method used here can be also extended to many other nonlinear partial differential equations.     

Explicit Solutions for Generalized （2＋1）-Dimensional Nonlinear Zakharov-Kuznetsov Equation

The exact solutions of the generalized （2＋1）-dimensional nonlinear Zakharov-Kuznetsov （Z-K） equation are explored by the method of the improved generalized auxiliary differential equation. Many explicit analytic solutions of the Z-K equation are obtained. The methods used to solve the Z-K equation can be employed in further work to establish new solutions for other 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.     

Auto-Backlund Transformation and Analytic Solutions of （2＋1）-Dimensional Boussinesq Equation

Using the truncated Painleve expansion, symbolic computation, and direct integration technique, we study analytic solutions of （2＋1）-dimensional Boussinesq equation. An auto-Backlund transformation and a number of exact solutions of this equation have been found. The set of solutions include solitary wave solutions, solitoff solutions, and periodic solutions in terms of elliptic Jacobi functions and Weierstrass ＆ function. Some of them are novel.     

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.     

New Compacton-Like and Solitary Pattern-Like Solutions of （2＋1）-Dimensional Generalization of Modified KdV Equation

Recently some （1＋1）-dimensional nonlinear wave equations with linearly dispersive terms were shown to possess compacton-like and solitary pattern-like solutions. In this paper, with the aid of Maple, new solutions of （2＋1）- dimensional generalization of mKdV equation, which is of only linearly dispersive terms, are investigated using three new transformations. As a consequence, it is shown that this （2＋1）-dimensional equation also possesses new compacton-like solutions and solitary pattern-like solutions.     

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.     

A New Rational Algebraic Approach to Find Exact Analytical Solutions to a （2＋1）-Dimensional System

In this paper, we present a new rational algebraic approach to uniformly construct a series of exact analytical solutions for nonlinear partial differential equations. Compared with most existing tanh methods and other sophisticated methods, the proposed method not only recovers some known solutions, but also finds some new and general solutions. The solutions obtained in this paper include rational form triangular periodic wave solutions, solitary wave solutions, and elliptic doubly periodic wave solutions. The efficiency of the method can be demonstrated on （2＋1）-dimensional dispersive long-wave 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.     

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.