Kawahara equation and modified Kawahara equation with time dependent coefficients: symmetry analysis and generalized ‐expansion method

In this paper, variable coefficients Kawahara equation (VCKE) and variable coefficients modified Kawahara equation (VCMKE), which arise in modeling of various physical phenomena, are studied by Lie group analysis. The similarity reductions and exact solutions are derived by determining the complete sets of point symmetries of these equations. Moreover, some exact analytic solutions are considered by the power series method. Further, a generalized ‐expansion method is applied to VCKE and VCMKE for constructing some new exact solutions. As a result, hyperbolic function solutions, trigonometric function solutions and some rational function solutions with parameters are furnished. Copyright © 2012 John Wiley & Sons, Ltd.     

Symmetries and conservation laws of Kadomtsev-Pogutse equations

Kadomtsev-Pogutse equations are of great interest from the viewpoint of the theory of symmetries and conservation laws and, in particular, enable us to demonstrate their potentials in action. This paper presents, firstly, the results of computations of symmetries and conservation laws for these equations and the methods of obtaining these results. Apparently, all the local symmetries and conservation laws admitted by the considered equations are exhausted by those enumerated in this paper. Secondly, we point out some reductions of Kadomtsev-Pogutse equations to more simpler forms which have less independent variables and which, in some cases, allow us to construct exact solutions. Finally, the technique of solution deformation by symmetries and their physical interpretation are demonstrated.     

Nonlocal symmetries and exact solutions of the (2+1)-dimensional generalized variable coefficient shallow water wave equation

In this paper, using the standard truncated Painlevé analysis, the Schwartzian equation of (2+1)-dimensional generalized variable coefficient shallow water wave (SWW) equation is obtained. With the help of Lax pairs, nonlocal symmetries of the SWW equation are constructed which be localized by a complicated calculation process. Furthermore, using the Lie point symmetries of the closed system and Schwartzian equation, some exact interaction solutions are obtained, such as soliton–cnoidal wave solutions. Corresponding 2D and 3D figures are placed to illustrate dynamic behavior of the generalized variable coefficient SWW equation.     

Complete Group Classification and Exact Solutions to the Generalized Short Pulse Equation

In this paper, the complete group classification is performed on the generalized short pulse equation, which includes a lot of important nonlinear wave equations as its special cases. In the sense of geometric symmetry, all of the vector fields of the equation are obtained in terms of the arbitrary functions. Then, the symmetry reductions and exact solutions to the equations are investigated. Especially, we develop the analytic power series method for constructing the exact power series solutions to the short pulse types of equations.     

Symmetry analysis and exact solutions of some Ostrovsky equations

We apply the classical Lie method and the nonclassical method to a generalized Ostrovsky equation (GOE) and to the integrable Vakhnenko equation (VE), which Vakhnenko and Parkes proved to be equivalent to the reduced Ostrovsky equation. Using a simple nonlinear ordinary differential equation, we find that for some polynomials of velocity, the GOE has abundant exact solutions expressible in terms of Jacobi elliptic functions and consequently has many solutions in the form of periodic waves, solitary waves, compactons, etc. The nonclassical method applied to the associated potential system for the VE yields solutions that arise from neither nonclassical symmetries of the VE nor potential symmetries. Some of these equations have interesting behavior such as “nonlinear superposition.”     

Painlevé analysis, Lie symmetries and exact solutions for (2+1)-dimensional variable coefficients Broer-Kaup equations

A (2+1) dimensional Broer-Kaup system which is obtained from the constraints of the KP equation is of importance in mathematical physics field. In this paper, the Painlevé analysis of (2+1)-variable coefficients Broer-Kaup (VCBK) equation is performed by the Weiss-Kruskal approach to check the Painlevé property. Similarity reductions of the VCBK equation to one-dimensional partial differential equations including Burger’s equation are investigated by the Lie classical method. The Lie group formalism is applied again on one of the investigated partial differential equation to derive symmetries, and the ordinary differential equations deduced from the optimal system of subalgebras are further studied and some exact solutions are obtained.     

Exact solutions to nonlinear Schrödinger equation with variable coefficients

According to Ma-Fuchsseiter’s idea, a trial equation method was proposed to find the exact envelop traveling wave solutions to some nonlinear differential equations with variable coefficients. As an application, combining with the complete discrimination system for polynomial, some exact envelop traveling wave solutions to Schrödinger equation with variable coefficients were obtained. At the same time, the physical meanings of the obtained solutions are discussed, and the problem needed to further study is pointed out.     

Conditional Lie–Bäcklund symmetries and functionally generalized separable solutions to the generalized porous medium equations with source

The functionally generalized separable solutions of the generalized porous medium equations with power law and exponential diffusivity are studied by using the conditional Lie–Bäcklund symmetry method. The variant forms of the considered equations, which admit the linear conditional Lie–Bäcklund symmetries, are identified. A number of examples are considered and some exact solutions, defined on the polynomial, trigonometric and exponential invariant subspaces determined by the linear conditional Lie–Bäcklund symmetries, are constructed.     

Lie group classification and invariant solutions of mKdV equation with time-dependent coefficients

This paper studies the modified Korteweg–de Vries equation with time variable coefficients of the damping and dispersion using Lie symmetry methods. We carry out Lie group classification with respect to the time-dependent coefficients. Lie point symmetries admitted by the mKdV equation for various forms for the time variable coefficients are obtained. The optimal system of one-dimensional subalgebras of the Lie symmetry algebras are determined. These are then used to determine exact group-invariant solutions, including soliton solutions, and symmetry reductions for some special forms of the equations.     

Generalized procedure of separation of variables and reduction of nonlinear wave equations

We propose a generalized procedure of separation of variables for nonlinear wave equations and construct broad classes of exact solutions of these equations that cannot be obtained by the classical Lie method and the method of conditional symmetries.     

Nonlocal symmetries and formulae for generation of solutions for a class of diffusion-convection equations

New formulae of nonlocal nonlinear superposition and generation of solutions are proposed for nonlinear diffusion-convection equations which are linearizable or are invariant with respect to a generalized hodograph transformation or connected by this transformation. We study in what particular ways additional Lie symmetries of diffusion-convection equations induce nonlocal symmetries of equations obtained from the initial ones by nonlocal transformations. The formulae derived are used for the construction of exact solutions.     

New periodic wave solutions for nonlinear evolution equations with variable coefficients via mapping method

An extended mapping method with a computerized symbolic computation is used for constructing a new exact travelling wave solutions for nonlinear evolution equations arising in physics, namely, generalized Zakharov Kuznetsov equation with variable coefficients. As a result, many exact travelling wave solutions are obtained which include new periodic wave solution, trigonometric function solutions and rational solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations with variable coefficients arising in mathematical physics. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source

A new approach to group classification problems and more general investigations on transformational properties of classes of differential equations is proposed. It is based on mappings between classes of differential equations, generated by families of point transformations. A class of variable coefficient (1+1)-dimensional semilinear reaction–diffusion equations of the general form f(x)u _t =(g(x)u _x ) _x +h(x)u ^m (m≠0,1) is studied from the symmetry point of view in the framework of the approach proposed. The singular subclass of the equations with m=2 is singled out. The group classifications of the entire class, the singular subclass and their images are performed with respect to both the corresponding (generalized extended) equivalence groups and all point transformations. The set of admissible transformations of the imaged class is exhaustively described in the general case m≠2. The procedure of classification of nonclassical symmetries, which involves mappings between classes of differential equations, is discussed. Wide families of new exact solutions are also constructed for equations from the classes under consideration by the classical method of Lie reductions and by generation of new solutions from known ones for other equations with point transformations of different kinds (such as additional equivalence transformations and mappings between classes of equations).     

A rational generalization of Fan’s method and its application to generalized shallow water wave equations

In this paper we employ a rational expansion to generalize Fan’s method for exact travelling wave solutions for nonlinear partial differential equations (PDEs). To verify the reliability of the proposed method, the generalized shallow water wave (GSWW) equation has been investigated as an example. Kinds of new exact travelling wave solutions of a rational form have been obtained. This indicates that the proposed method provides a more general result for exact solution of nonlinear equations.     

Numerical solutions of boundary value problems for variable coefficient generalized KdV equations using Lie symmetries

The exhaustive group classification of a class of variable coefficient generalized KdV equations is presented, which completes and enhances results existing in the literature. Lie symmetries are used for solving an initial and boundary value problem for certain subclasses of the above class. Namely, the found Lie symmetries are applied in order to reduce the initial and boundary value problem for the generalized KdV equations (which are PDEs) to an initial value problem for nonlinear third-order ODEs. The latter problem is solved numerically using the finite difference method. Numerical solutions are computed and the vast parameter space is studied.     

Symmetries for a family of Boussinesq equations with nonlinear dispersion

In this paper, we make a full analysis of a family of Boussinesq equations which include nonlinear dispersion by using the classical Lie method of infinitesimals. We consider travelling wave reductions and we present some explicit solutions: solitons and compactons.For this family, we derive nonclassical and potential symmetries. We prove that the nonclassical method applied to these equations leads to new symmetries, which cannot be obtained by Lie classical method. We write the equations in a conserved form and we obtain a new class of nonlocal symmetries. We also obtain some Type-II hidden symmetries of a Boussinesq equation.     

多孔介质方程的广义条件对称和精确解

运用广义条件对称方法对径向对称的多孔介质方程进行了对称约化.确定了允许二阶广义条件对称的方程形式,并给出了方程相应的不变解.     

(2+1)维广义Burgers 方程的Lie点对称, 相似约化和精确解

讨论了(2+1)维广义Burgers方程.通过Lie群方法求出了该方程的李点对称,并利用李点对称将方程进行相似约化,求出了(2+1)维广义Burgers方程的几种精确解.该方法可以用于研究更高阶的偏微分方程.     

Moving planes, moving spheres, and a priori estimates

In this paper, we use the method of moving planes and the method of moving spheres to obtain a priori estimates for the solutions of semi-linear elliptic equations.Using the ‘method of moving planes’, we establish a sharper estimate on the solutions for prescribing scalar curvature equations with indefinite curvature functions and thus generalized a resent result of Lin.Applying ‘the method of moving spheres’, we give a different proof for a well-known sup+inf inequality established by Brezis, Li and Shafrir.