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.     

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.     

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.     

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.     

New Baecklund Transformation and New Exact Solutions to （2＋1）-Dimensional KdV Equation

A new Baecklund transformation for (2 1)-dimensional KdV equation is first obtained by using homogeneous balance method. And making use of the Baecklund transformation and choosing a special seed solution, we get special types of solitary wave solutions. Finally a general variable separation solution containing two arbitrary functions is constructed, from which abundant localized coherent structures of the equation in question can be induced.     

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.     

Exact Periodic-Wave Solutions for （2＋1）-Dimensional Boussinesq Equation and（3＋1）-Dimensional KP Equation

The (2 1)-dimensional Boussinesq equation and (3 1)-dimensional KP equation are studied by using the extended Jacobi elliptic-function method. The exact periodic-wave solutions for the two equations are obtained.     

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.     

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.     

A Series of Exact Solutions of （2＋l）-Dimensional CDGKS Equation

An algebraic method with symbolic computation is devised to uniformly construct a series of exact solutions of the （2＋1）-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawda equation. The solutions obtained in this paper include solitary wave solutions, rational solutions, triangular periodic solutions, Jacobi and Weierstrass doubly periodic solutions. Among them, the Jacobi periodic solutions exactly degenerate to the solutions at a certain limit condition. Compared with most existing tanh method, the method used here can give new and more general solutions. More importantly, this method provides a guldeline to classifj, the various types of the solution according to some parameters.     

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.     

Periodic Wave Solutions for （2＋1）-Dimensional Boussinesq Equation and （3＋ 1）-Dimensional Kadomtsev-Petviashvili Equation

For describing various complex nonlinear phenomena in the realistic world, the higher-dimensional nonlinear evolution equations appear more attractive in many fields of physical and engineering sciences. In this paper, by virtue of the Hirota bilinear method and Riemann theta functions, the periodic wave solutions for the （2＋1）-dimensional Boussinesq equation and （3＋1）-dimensional Kadomtsev Petviashvili （KP） equation are obtained. Furthermore, it is shown that the known soliton solutions for the two equations can be reduced from the periodic wave solutions.     

New Multiple Soliton-like Solutions to （3＋1）-Dimensional Burgers Equation with Variable Coefficients

A new generalized tanh function method is used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the Riccati equation, which has more new solutions. More new multiple soliton-like solutions are obtained for the (3 1)-dimensional Burgers equation with variable coefficients.     

New Solitary Solutions of （2＋1）-Dimensional Variable Coefficient Nonlinear Schrodinger Equation with an External Potential

By a series of transformations, the （2＋1）-dimensional variable coefficient nonlinear Schr？dinger equation can turn to the Klein-Gordon equation. Many new double travelling wave solutions of the Klein-Gordon equation are obtained. Thus, the new solitary solutions of the variable coefficient nonlinear Schr？inger equation with an external potential can be found.     

Exact Solutions of （2＋1）-Dimensional Euler Equation Found by Weak Darboux Transformation

The weak Darboux transformation of the （2＋1） dimensional Euler equation is used to find its exact solutions. Starting from a constant velocity field solution, a set of quite general periodic wave solutions such as the Rossby waves can be simply obtained from the weak Darboux transformation with zero spectral parameters. The constant vorticity seed solution is utilized to generate Bessel waves.     

New Exact Solutions to （2＋1）-Dimensional Variable Coefficients Broer-Kaup Equations

In this paper, using the variable coefficient generalized projected Rieatti equation expansion method, we present explicit solutions of the （2＋1）-dimensional variable coefficients Broer-Kaup （VCBK） equations. These solutions include Weierstrass function solution, solitary wave solutions, soliton-like solutions and trigonometric function solutions. Among these solutions, some are found for the first time. Because of the three or four arbitrary functions, rich localized excitations can be found.     

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.     

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.     

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.     

New Exact Solutions and Dynamics in （3＋1）-Dimensional Gross-Pitaevskii Equation with Repulsive Harmonic Potential

Using an improved homogeneous balance principle and an F-expansion technique, we construct the new exact periodic traveling wave solutions to the （3＋1）-dimensional Gross-Pitaevskii equation with repulsive harmonic potential. In the limit cases, the solitary wave solutions are obtained as well. We also investigate the dynamical evolution of the solitons with a time-dependent complicated potential.