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.     

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 reduction and exact solutions of the (3+1)-dimensional Zakharovben Kuznetsov equation

By means of the classical method, we investigate the (3+1)-dimensional Zakharov-Kuznetsov equation. The symmetry group of the (3+1)-dimensional Zakharov-Kuznetsov equation is studied first and the theorem of group invariant solutions is constructed. Then using the associated vector fields of the obtained symmetry, we give the one-, two-, and three-parameter optimal systems of group-invariant solutions. Based on the optimal system, we derive the reductions and some new solutions of the (3+1)-dimensional Zakharov-Kuznetsov equation.     

Some New Lie Symmetry Groups of Differential-Difference Equations Obtained from a Simple Direct Method

In this paper, based on the symbolic computing system Maple, the direct method for Lie symmetry groups presented by Sen-Yue Lou [J. Phys. A: Math. Gen. 38 (2005) L129] is extended from the continuous differential equations to the differential-difference equations. With the extended method, we study the well-known differential-difference KP equation, KZ equation and (2+1)-dimensional ANNV system, and both the Lie point symmetry groups and the non-Lie symmetry groups are obtained.     

一类高阶非线性波方程的李群分析、最优系统、精确解和守恒律

本文运用李群分析的方法研究了一类高阶非线性波方程,得到了五阶非线性波方程的对称以及方程的最优系统,进而运用幂级数的方法,求得了方程的精确幂级数解.最后,给出了五阶非线性波方程的一些守恒律.     

Lie symmetry analysis,new group invariant for the (3 + 1)-dimensional and variable coefficients for liquids with gas bubbles models

The explored solutions described some different solutions as, Lump soliton, a solitary wave and exponential solutions. These solutions are investigated through some new Lie infinitesimals for the (3 + 1) dimensional variable coefficients Kudryashov-Sinelshchikov (VCKS). We used the fourth prolongation to investigate fifteen cases of Lie vectors. In each case, there is an infinite number of possibilities of vectors due to the unknown arbitrary functions and the variable coefficients for the considered model. We selected one case and examined the commutative product between multi unknown Lie infinitesimals for the (3 + 1) dimensional (VCKS) equation and this complicated process resulted from some new Lie vectors. The commutative product generates a system of nonlinear ODEs which had been solved manually. Through three stages of Lie symmetry reduction using the equivalence transformation, (VCKS) equation is reduced to solvable nonlinear ODEs using various combinations of optimal Lie vectors. By solving these ODEs, we investigate new analytical solutions for these ODEs. Back substituting to the original variables generates new solutions for (VCKS). Some selected solutions are illustrated through three-dimensional plots.     

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.     

Exotic interactions between solitons of the （2＋1）-dimensional asymmetric Nizhnik-Novikov-Veselov system

Starting from the extended tanh-function method (ETM) based on the mapping method, the variable separation solutions of the (2+1)-dimensional asymmetric Nizhnik--Novikov--Veselov (ANNV) system are derived. By further study, we find that these variable separation solutions are seemingly independent of but actually dependent on each other. Based on the variable separation solution and by choosing appropriate functions, some novel and interesting interactions between special solitons, such as bell-like compacton, peakon-like compacton and compacton-like semi-foldon, are investigated.     

Symmetry Analysis of Two Types of (2+1)-Dimensional NonlinearKlein-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.     

New Exact Solutions and Conservation Laws to (3+1)-Dimensional Potential-YTSF Equation

Using the modified CK's direct method, we build the relationship between new solutions and old ones and find some new exact solutions to the (3+1)-dimensional potential-YTSF equation. Based on the invariant group theory, Lie point symmetry groups and Lie symmetries of the (3+1)-dimensional potential-YTSF equation are obtained. We also get conservation laws of the equation with the given Lie symmetry.     

Symmetry Reductions and Explicit Solutions for a Generalized Zakharov-Kuznetsov Equation

Two types of symmetry of a generalized Zakharov-Kuznetsov equation are obtained via a direct symmetry method. By selecting suitable parameters occurring in the symmetries, we also find some symmetry reductions and new explicit solutions of the generalized Zakharov-Kuznetsov equation.     

A series of new double periodic solutions to a (2+1)-dimensional asymmetric Nizhnik－Novikov－Veselov equation

By means of a new general ans?tz and with the aid of symbolic computation, a new algebraic method named Jacobi elliptic function rational expansion is devised to uniformly construct a series of new double periodic solutions to (2+1)-dimensional asymmetric Nizhnik－Novikov－Veselov (ANNV) equation in terms of rational Jacobi elliptic function.     

Symmetry groups and spiral wave solution of a wave propagation equation

We study a third-order nonlinear evolution equation, which can be transformed to the modified KdV equation, using the Lie symmetry method. The Lie point symmetries and the one-dimensional optimal system of the symmetry algebras are determined. Those symmetries are some types of nonlocal symmetries or hidden symmetries of the modified KdV equation. The group-invariant solutions, particularly the travelling wave and spiral wave solutions, are discussed in detail, and a type of spiral wave solution which is smooth in the origin is obtained.     

Symmetry properties,conservation laws and exact solutions of time-fractional irrigation equation

In this paper, we deal with the complete algebra of Lie point symmetries for the generalized model of an irrigation system of fractional order. By means of Lie symmetry method, the vector fields has been investigated which are utilized for obtaining the conservation laws of equation. In addition, through the sub-equation method, we construct some exact solutions for the considered equation by reducing the fractional partial differential equation to a ordinary fractional differential equation.     

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.     

Dromion and other exact solutions of (2+1)-dimensional higher-order Broer-Kaup system

Using extended homogenous balance method, we obtain Bäcklund transformation (BT) and a linear partial differential equation of higher-order Broer-Kaup (HBK) system. As a result, multisoliton and single soliton and other exact solutions of (2+1)-dimensional HBK system are given. By analyzing single soliton solution, we get some dromion solutions.     

A nonlocal Burgers equation in atmospheric dynamical system and its exact solutions

From a two-vortex interaction model in atmospheric and oceanic systems, a nonlocal counterpart with shifted parity and delayed time reversal is derived by a simple AB reduction. To obtain some approximate analytic solutions of this nonlocal system, the multi-scale expansion method is applied to get an AB-Burgers system. Various exact solutions of the AB-Burgers equation, including elliptic periodic waves, kink waves and solitary waves, are obtained and shown graphically.To show the applications of these solutions in describing correlated events, a simple approximate solution for the two-vortex interaction model is given to show two correlated dipole blocking events at two different places. Furthermore, symmetry reduction solutions of the nonlocal AB-Burgers equation are also given by using the standard Lie symmetry method.     

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 Schrödinger (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.     

Nonlocal symmetry,optimal systems,and explicit solutions of the mKdV equation

The nonlocal symmetry of the mKdV equation is obtained from the known Lax pair; it is successfully localized to Lie point symmetries in the enlarged space by introducing suitable auxiliary dependent variables. For the closed prolongation of the nonlocal symmetry, the details of the construction for a one-dimensional optimal system are presented. Furthermore, using the associated vector fields of the obtained symmetry, we give the reductions by the one-dimensional sub-algebras and the explicit analytic interaction solutions between cnoidal waves and kink solitary waves, which provide a way to study the interactions among these types of ocean waves. For some of the interesting solutions, the figures are given to show their properties.     

广义(3+1)维Zakharov-Kuznetsov方程的对称约化、精确解和守恒律

运用李群分析,得到了广义(3+1)维Zakharov-Kuznetsov(ZK)方程的对称及约化方程,结合齐次平衡原理,试探函数法和指数函数法得到了该方程的群不变解和新精确解,包括冲击波解、孤立波解等.进一步给出了广义(3+1)维ZK方程的伴随方程和守恒律.