A Note on Nonclassical Symmetries of a Class of Nonlinear Partial Differential Equations and Compatibility

The nonclassical symmetries of a class of nonlinear partial differential equations obtained by the compatibilitymethod is investigated.We show the nonclassical symmetries obtained in [J.Math.Anal.Appl.289 (2004) 55,J.Math.Anal.Appl.311 (2005) 479] are not all the nonclassical symmetries.Based on a new assume on the form ofinvariant surface condition,all the nonclassical symmetries for a class of nonlinear partial differential equations can beobtained through the compatibility method.The nonlinear Klein-Gordon equation and the Cahn-Hilliard equations allserve as examples showing the compatibility method leads quickly and easily to the determining equations for their allnonclassical symmetries for two equations.

Binary Bell Polynomials,Bilinear Approach to Exact Periodic Wave Solutions of(2+l)-Dimensional Nonlinear Evolution Equations

In the present letter,we get the appropriate bilinear forms of(2 + l)-dimensionaI KdV equation,extended (2+1)-dimensional shallow water wave equation and(2 +1)-dimensional Sawada-Kotera equation in a quick and natural manner,namely by appling the binary Bell polynomials.Then the Hirota direct method and Riemann theta function are combined to construct the periodic wave solutions of the three types nonlinear evolution equations.And the corresponding figures of the periodic wave solutions are given.Furthermore,the asymptotic properties of the periodic wave solutions indicate that the soliton solutions can be derived from the periodic wave solutions.

Full Symmetry Groups and Similar Reductions of a (2+1)-Dimensional Resonant Davey-Stewartson System

把古典谎言对称方法用于( 2+1 )维的反响的 DaveyStewartson 系统由唐介绍了[唐克西等，混乱， Solitons 和分数维图形 42 ( 2007 ) 2707 ]，有 Kac-Moody-Virasoro 类型谎言代数学的更一般的无限的维的谎言对称被获得，它包含 t 的四任意的功能。由一个简单直接方法，或者，包括的完整的对称组躺着对称组和非谎言对称组直地被获得。这样，相关谎言代数学能被一个更简单的限制过程容易发现。经由解决典型方程，最后，一般类似的减小的三种类型被导出。

Adomian Decomposition Method and Padé Approximants for Nonlinear Differential-Difference Equations

有 Pad é approximants 的联合 Adomian 分解方法(ADM ) ，我们解决二个微分差别的方程(DDE ) ：相对论的 Toda 格子方程和修改 Volterra 格子方程。在符号的计算枫树的帮助下， ADM-Pad é技术获得的结果与独自使用 ADM 获得的那些相比。数字结果比使用 ADM 证明 ADM-Pad é技术在集中的更大的领域与更快的集中率和更高的精确性和亲戚给近似答案。

Extended symmetry transformation of (3+1)-dimensional generalized nonlinear Schrdinger equation with variable coefficients

In this paper, the extended symmetry transformation of （3＋1）-dimensional （3D） generalized nonlinear Schrodinger （NLS） equations with variable coefficients is investigated by using the extended symmetry approach and symbolic computation. Then based on the extended symmetry, some 3D variable coefficient NLS equations are reduced to other variable coefficient NLS equations or the constant coefficient 3D NLS equation. By using these symmetry transformations, abundant exact solutions of some 3D NLS equations with distributed dispersion, nonlinearity, and gain or loss are obtained from the constant coefficient 3D NLS equation.

Symmetry and general symmetry groups of the coupled Kadomtsev-Petviashvili equation

In this paper, the Lie symmetry algebra of the coupled Kadomtsev-Petviashvili （cKP） equation is obtained by the classical Lie group method and this algebra is shown to have a Kac-Moody-Virasoro loop algebra structure. Then the general symmetry groups of the cKP equation is also obtained by the symmetry group direct method which is proposed by Lou et alo From the general symmetry groups, the Lie symmetry group can be recovered and a group of discrete transformations can be derived simultaneously. Lastly, from a known simple solution of the cKP equation, we can easily obtain two new solutions by the general symmetry groups.

Symmetry analysis and explicit solutions of the (3+1)-dimensional baroclinic potential vorticity equation

This paper investigates an important high-dimensional model in the atmospheric and oceanic dynamics-（3＋1）- dimensional nonlinear baroclinic potential vorticity equation by the classical Lie group method. Its symmetry algebra, symmetry group and group-invariant solutions are analysed. Otherwise, some exact explicit solutions are obtained from the corresponding （2＋1）-dimensional equation, the inviscid barotropic nondivergent vorticy equation. To show the properties and characters of these solutions, some plots as well as their possible physical meanings of the atmospheric circulation are given out.

Bcklund Transformations and Explicit Solutions of (2+1)-Dimensional Barotropic and Quasi-Geostrophic Potential Vorticity Equation

By the Bcklund transformation method, an important (2+1)-dimensional nonlinear barotropic and quasigeostrophicpotential vorticity (BQGPV) equation is investigated. Some simple special Bcklund transformation theoremsare proposed and used to get explicit solutions of the BQGPV equation. Futhermore, all solutions of a secondorder linear ordinary differential equation including an arbitrary function can be used to construct explicit solutions ofthe (2+1)-dimensional BQGPV equation. Some figures are also given out to describe these solutions.

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 groupsof the well-known (2+1)-dimensional Generalized Sasa-Satsuma (GSS) equation. Second, based on a modified directmethod proposed by Lou [J. Phys. A: Math. Gen. 38 (2005) L129], more general symmetry groups are obtained andthe relationship between the new solution and known solution is set up. At the same time, the Lie symmetry groupsobtained are only special cases of the more general symmetry groups. At last, some exact solutions of GSS equationsare constructed by the relationship obtained in the paper between the new solution and known solution.

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 nonlinearKlein-Gorden equation.For the wave equation,we give out its symmetry group analysis in detail.For the secondtype of the(2+1)-dimensional nonlinear Klein-Gorden equation,an optimal system of its one-dimensional subalgebrasis constructed and some corresponding two-dimensional symmetry reductions are obtained.

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 mKd V 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.

Numerical Solutions of a Class of Nonlinear Evolution Equations with Nonlinear Term of Any Order

<Abstract>In this paper,the Adomian decomposition method is developed for the numerical solutions of a class of nonlinear evolution equations with nonlinear term of any order,u_(tt)+au_(xx)+bu + cu^p+ du^(2p-1) = 0 which contains some important famous equations.When setting the initial conditions in different forms,some new generalized numerical solutions:numerical hyperbolic solutions,numerical doubly periodic solutions are obtained.The numerical solutions are compared with exact solutions.The scheme is tested by choosing different values of p,positive and negative,integer and fraction,to illustrate the efficiency of the ADM method and the generalization of the solutions.

Function Projective Synchronization of Discrete-Time Chaotic and Hyperchaotic Systems Using Backstepping Method

In this paper,a function projective synchronization scheme is developed to investigate the function projec-tive synchronization between the discrete-time driven chaotic system and the discrete-time response chaotic system.Withthe aid of symbolic-numeric computation,we use the scheme to study the function projective synchronization between2D Lorenz discrete-time system and Hénon discrete-time system,as well as that between 3D discrete-time hyperchaoticsystem and Hénon-like map via three scalar controllers,respectively.Moreover numerical simulations are used to verifythe effectiveness of the proposed scheme.

Numerical Solutions of a New Type of Fractional Coupled Nonlinear Equations

在这篇论文，我们调查部分联合非线性的方程的一种新类型。由介绍满足 Caputo 的定义的部分衍生物，我们直接扩大 Adomian 分解方法的应用到新系统。作为结果，在枫树的帮助下，现实主义、会聚很快，系列答案与容易可计算出来的部件被获得。二个著名部分联合例子：KdV 和 mKdV 方程，被用来说明建议方法的效率和精确性。

Dynamics of solitons of the generalized(3+1)-dimensional nonlinear Schr(o|¨)dinger equation with distributed coefficients

We present three families of soliton solutions to the generalized （3＋l）-dimensional nonlinear Schrodinger equation with distributed coefficients. We investigate the dynamics of these solitons in nonlinear optics with some selected parameters. Different shapes of bright solitons, a train of bright solitons and dark solitons are observed. The obtained results may raise the possibilities of relevant experiments and potential applications.

Novel exact solutions of coupled nonlinear Schro¨dinger equations with time–space modulation

We construct various novel exact solutions of two coupled dynamical nonlinear Schrōdinger equations. Based on the similarity transformation, we reduce the coupled nonlinear Schrōdinger equations with time-and space-dependent potentials, nonlinearities, and gain or loss to the coupled dynamical nonlinear Schrrdinger equations. Some special types of non-travelling wave solutions, such as periodic, resonant, and quasiperiodically oscillating solitons, are used to exhibit the wave propagations by choosing some arbitrary functions. Our results show that the number of the localized wave of one component is always twice that of the other one. In addition, the stability analysis of the solutions is discussed numerically.

Complexiton solutions of the （2＋1）-dimensional dispersive long wave equation

In this pager a pure algebraic method implemented in a computer algebraic system, named multiple Riccati equations rational expansion method, is presented to construct a novel class of complexiton solutions to integrable equations and nonintegrable equations. By solving the （2＋1）-dimensional dispersive long wave equation, it obtains many new types of complexiton solutions such as various combination of trigonometric periodic and hyperbolic function solutions, various combination of trigonometric periodic and rational function solutions, various combination of hyperbolic and rationai function solutions, etc.

Exact analytical solutions of three-dimensional Gross-Pitaevskii equation with time-space modulation

By the generalized sub-equation expansion method and symbolic computation, this paper investigates the （3＋1）dimensional Gross-Pitaevskii equation with time-and space-dependent potential, time-dependent nonlinearity, and gain or loss. As a result, rich exact analytical solutions are obtained, which include bright and dark solitons, Jacobi elliptic function solutions and Weierstrass elliptic function solutions. With computer simulation, the main evolution features of some of these solutions are shown by some figures. Nonlinear dynamics of a soliton pulse is also investigated under the different regimes of soliton management.

New Complexiton Solutions of （1＋1）-Dimensional Dispersive Long Wave Equation

借助于有在有限合理扩大方法的部件的不同参数 assubequation 的二个不同 Riccati 方程，的新 complexiton 解决方案( 1+1 )维的散长波浪方程成功地被构造，它包括三角法的周期、夸张的函数解决方案的各种各样的联合，三角法的周期、合理的函数解决方案的各种各样的联合，和夸张、合理的函数解决方案的各种各样的联合。