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.     

Darboux Transformations and N-soliton Solutions of Two （2＋1）-Dimensional Nonlinear Equations

Two Darboux transformations of the （2＋1）-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawaka （ CDGKS） equation and （2＋1）-dimensional modified Korteweg-de Vries （mKdV） equation are constructed through the Darboux matrix method, respectively. N-soliton solutions of these two equations are presented by applying the Darboux trans- formations N times. The right-going bright single-soliton solution and interactions of two and three-soliton overtaking collisions of the （2＋1）-dimensional CDGKS equation are studied. By choosing different seed solutions, the right-going bright and left-going dark single-soliton solutions, the interactions of two and three-soliton overtaking collisions, and kink soliton solutions of the （2＋1）-dimensional mKdV equation are investigated. The results can be used to illustrate the interactions of water waves in shallow water.     

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.     

Higher-Dimensional Integrable Systems Induced by Motions of Curves in Affine Geometries

We discuss the motions of curves by introducing an extra spatial variable or equivalently, moving surfaces in arffine geometries. It is shown that the 2 ＋1-dimensional breaking soliton equation and a 2 ＋ 1-dimensional nonlinear evolution equation regarded as a generalization to the 1 ＋ 1-dimensional KdV equation arise from such motions.     

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.     

Explicit Solutions of （2A-1）-Dimensional Canonical Generalized KP,KdV, and （2A-1）-Dimensional Burgers Equations with Variable Coefficients

In this paper, using the generalized （G1/G）-expansion method and the auxiliary differential equation method, we discuss the （2＋1）-dimensional canonical generalized KP （CGKP）, KdV, and （2＋1）-dimensional Burgers equations with variable coetticients. Many exact solutions of the equations are obtained in terms of elliptic functions, hyperbolic functions, trigonometric functions, and rational functions.     

Propagation of light in （2＋1）-dimensional nonlinear optical media with spatially inhomogeneous nonlinearities

The （2＋1）-dimensional nonlinear SchrSdinger （NLS） equation with spatially inhomogeneous nonlinearities is investigated, which describes propagation of light in （2＋1）-dimensional nonlinear optical media with inhomogeneous nonlinearities. New types of optical modes and nonlinear effects in optical media are presented numerically. The results reveal that the regular split of beam can be obtained in （2＋1）-dimensional nonlinear optical media with inhomogeneous nonlinearities, by adjusting the guiding parameter. Furthermore, the stability of beam regular split is discussed numerically, and the results reveal that the beam regular split is stable to the finite initial perturbations.     

Invariance of Painlev6 property for some reduced （1 ＋ 1）-dimensional equations

We study the Painlevé property of the （1＋1）-dimensional equations arising from the symmetry reduction for the （2＋1）- dimensional ones. Firstly, we derive the similarity reduction of the （2＋1）-dimensional potential Calogero-Bogoyavlenskii- Schiff （CBS） equation and Konopelchenko-Dubrovsky （KD） equations with the optimal system of the admitted one-dimensional subalgebras. Secondly, by analyzing the reduced CBS, KD, and Burgers equations with Painlevé test, re-spectively, we find both the Painlevé integrability, and the number and location of resonance points are invariant, if the similarity variables include all of the independent variables.     

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.     

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.     

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.     

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 Multiple Soliton-like and Periodic Solutions for （2＋l）-Dimensional Canonical Generalized KP Equation with Variable Coefficients

In this paper, the generalized ranch function method is extended to （2＋1）-dimensianal canonical generalized KP （CGKP） equation with variable coetfficients. Taking advantage of the Riccati equation, many explicit exact solutions, which contain multiple soliton-like and periodic solutions, are obtained for the （2＋1）-dimensional OGKP equation with variable coetffcients.     

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.     

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.     

Explicit Solutions for a New （2＋1）-Dimensional Coupled mKdV Equation

An integrable （2＋1）-dimensional coupled mKdV equation is decomposed into two （1 ＋1）-dimensional soliton systems, which is produced from the compatible condition of three spectral problems. With the help of decomposition and the Darboux transformation of two （1＋1）-dimensional soliton systems, some interesting explicit solutions of these soliton equations are obtained.     

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.     

Variable Separation Solutions in （1＋1）-Dimensional and （3＋1）-Dimensional Systems via Entangled Mapping Approach

In this paper, the entangled mapping approach （EMA） is applied to obtain variable separation solutions of （1＋1）-dimensional and （3＋1）-dimensional systems. By analysis, we firstly find that there also exists a common formula to describe suitable physical fields or potentials for these （1＋1）-dimensional models such as coupled integrable dispersionless （CID） and shallow water wave equations. Moreover, we find that the variable separation solution of the （3＋1）-dimensional Burgers system satisfies the completely same form as the universal quantity U1 in （2＋1）-dimensional systems. The only difference is that the function q is a solution of a constraint equation and p is an arbitrary function of three independent variables.     

Backlund Transformations and Explicit Solutions of （2d-1）-Dimensional Barotropic and Quasi-Geostrophic Potential Vorticity Equation

By the Backlund transformation method, an important （2＋1）-dimensional nonlinear barotropie and quasigeostrophic potential vorticity （BQGPV） equation is investigated. Some simple special Backlund transformation theorems are proposed and used to get explicit solutions of the BQGPV equation. Furthermore, all solutions of a second order linear ordinary differential equation including an arbitrary function can be used to construct explicit solutions of the （2＋1）-dimensional BQGPV equation. Some figures are also given out to describe these solutions.     

（2＋2）-Dimensional Discrete Soliton Equations and Integrable Coupling System

In this paper, we extend a （2＋2）-dimensional continuous zero curvature equation to （2＋2）-dimensional discrete zero curvature equation, then a new （2＋2）-dimensional cubic Volterra lattice hierarchy is obtained. Fhrthermore, the integrable coupling systems of the （2＋2）-dimensional cubic Volterra lattice hierarchy and the generalized Toda lattice soliton equations are presented by using a Lie algebraic system sl（4）.