On Lie symmetry analysis,conservation laws and solitary waves to a longitudinal wave motion equation

Under investigation in this work is a longitudinal wave motion equation, which describes the solitary waves propagation with dispersion caused by transverse Poisson’s effect in a magneto-electro-elastic circular rod. The Lie symmetry method is employed to study its vector fields and optimal systems, respectively. Furthermore, the symmetry reductions and eight families of soliton wave solutions of the equation are obtained on the basis of the optimal systems, including hyperbolic-type and trigonometric-type solutions. Two of reduced equations are Painlevé-like equations. Finally, by virtue of conservation law multiplier, the complete set of local conservation laws of the equation for the arbitrary constant coefficients is well constructed with a detailed derivation.     

BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS FOR THE GENERALIZED DODD-BULLOUGH-MIKHAILOV EQUATION

In this paper, the generalized Dodd-Bullough-Mikhailov equation is studied. The existence of periodic wave and unbounded wave solutions is proved by using the method of bifurcation theory of dynamical systems. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.Some exact explicit parametric representations of the above travelling solutions are obtained.     

Nonlocally related systems, linearization and nonlocal symmetries for the nonlinear wave equation

The nonlinear wave equation utt=(c²x(u)ux) arises in various physical applications. Ames et al. [W.F. Ames, R.J. Lohner, E. Adams, Group properties of utt=x[f(u)ux], Int. J. Nonlin. Mech. 16 (1981) 439-447] did the complete group classification for its admitted point symmetries with respect to the wave speed function c(u) and as a consequence constructed explicit invariant solutions for some specific cases. By considering conservation laws for arbitrary c(u), we find a tree of nonlocally related systems and subsystems which include related linear systems through hodograph transformations. We use existing work on such related linear systems to extend the known symmetry classification in [W.F. Ames, R.J. Lohner, E. Adams, Group properties of utt=x[f(u)ux], Int. J. Nonlin. Mech. 16 (1981) 439-447] to include nonlocal symmetries. Moreover, we find sets of c(u) for which such nonlinear wave equations admit further nonlocal symmetries and hence significantly further extend the group classification of the nonlinear wave equation.     

高阶Camassa—Holm方程的一类行波解

研究高阶Camassa-Holm方程的行波解,采用一种新的方法求解行波方程,获得了高阶Camassa-Holm方程的一类行波解．     

Benny-Luke 方程和 Phi-4方程的精确行波解

运用了tanh-coth展开法,得到了Benny-Luke方程和Phi-4方程的一些新的精确行波解。同时该方法也可用于其他的非线性演化方程。     

Bifurcations of exact travelling wave solutions for the generalized R-K-L equation

Using the method of dynamical systems for the the generalized Radhakrishnan, Kundu, Lakshmanan equation, the existence of soliton solutions, uncountably infinite many periodic wave solutions and unbounded wave solution are obtained. Exact explicit parametric representations of the above travelling solutions are given. To guarantee the existence of the above solutions, all parameter conditions are determined.     

The lie group and integrability of the Fisher type travelling wave equation

For the Fisher-type wave equation, which has two stable states and one unstable state, it is proved that only in two particular cases, the corresponding travelling wave equation admits a double parameter Lie group, and based on a method different to the traditional one, its two independent first integrals are given. It is proved further that in the two integrable cases, the different bounded and non-trivial travelling wave solutions, which are corresponding the invariant manifolds of the corresponding equation under the Lie transformation, can be expressed with elementary functions although they cannot be obtained directly from the two independent first integrals.     

NEW EXACT TRAVELLING WAVE SOLUTIONS TO THREE NONLINEAR EVOLUTION EQUATIONS

Based on the computerized symbolic computation, some new exact travelling wave solutions to three nonlinear evolution equations are explicitly obtained by replacing the tanhξ in tanh-function method with the solutions of a new auxiliary ordinary differential equation.     

New exact travelling wave solutions of two nonlinear physical models

In this paper, an improved tanh function method is used with a computerized symbolic computation for constructing new exact travelling wave solutions on two nonlinear physical models namely, the quantum Zakharov equations and the (2+1)-dimensional Broer–Kaup–Kupershmidt (BKK) system. The main idea of this method is to take full advantage of the Riccati equation which has more new solutions.The exact solutions are obtained which include new soliton-like solutions, trigonometric function solutions and rational solutions. The method is straightforward and concise, and its applications are promising.     

Group properties and conservation laws for nonlocal shallow water wave equation

Symmetry groups, symmetry reductions, optimal system, conservation laws and invariant solutions of the shallow water wave equation with nonlocal term are studied. First, Lie symmetries based on the invariance criterion for nonlocal equations and the solution approach for nonlocal determining equations are found and then the reduced equations and optimal system are obtained. Finally, new conservation laws are generated and some similarity solutions for symmetry reduction forms are discussed.     

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.     

Bifurcation studies on travelling wave solutions for an integrable nonlinear wave equation

By using the method of planar dynamical systems to an integrable nonlinear wave equation, the existence of periodic travelling wave, solitary wave and kink wave solutions is proved in the different parametric conditions. The phase portraits of the travelling wave system are given. It can be shown that the existence of singular curves in the travelling wave system is the reason why the travelling wave solutions lose their smoothness. Moreover, the so-called W/M-shaped solitary wave solutions are obtained.     

Global actions of Lie symmetries for the nonlinear heat equation

By restricting to a natural class of functions, we show that the Lie point symmetries of the nonlinear heat equation exponentiate to a global action of the corresponding Lie group. Remarkably, in most of the cases, the action turns out to be linear.     

Bifurcation of travelling wave solutions in a nonlinear variant of the RLW equation

By using the method of planar dynamical systems to a nonlinear variant of the regularized long-wave equation (RLW equation in short), the existence of smooth and non-smooth solitary wave (so called peakon and valleyon) and infinite many periodic wave solutions is shown. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. The formulas to compute the travelling waves are also educed. We notice that some results in [Wazwaz AM. Analytic study on nonlinear variants of the RLW and the PHI-four equations. Commun Nonlinear Sci Numer Simul, in press, doi:10.1016/j.cnsns.2005.03.001] are incorrect.     

Bifurcations of travelling wave solutions for the generalized (2+1)-dimensional Boussinesq-Kadomtsev-Petviashvili equation

By using the bifurcation theory of dynamical systems, we study the generalized (2+1)-dimensional Boussinesq-Kadomtsev-Petviashvili equation, the existence of solitary wave solutions, compacton solutions, periodic cusp wave solutions and uncountably infinite many smooth periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are determined.     

Exact solitary wave solutions of nonlinear wave equations

The hyperbolic function method for nonlinear wave equations is presented. In support of a computer algebra system, many exact solitary wave solutions of a class of nonlinear wave equations are obtained via the method. The method is based on the fact that the solitary wave solutions are essentially of a localized nature. Writing the solitary wave solutions of a nonlinear wave equation as the polynomials of hyperbolic functions, the nonlinear wave equation can be changed into a nonlinear system of algebraic equations. The system can be solved via Wu Elimination or Gr?bner base method. The exact solitary wave solutions of the nonlinear wave equation are obtained including many new exact solitary wave solutions.     

一个曲面波方程的精确解

Abstract. Some exact travelling wave solutions and rational travelling wave solutions of a sur-face wave equation in a convecting fluid are given in this paper.     

Bifurcations of travelling wave solutions for modified nonlinear dispersive phi-four equation

By using the bifurcation theory of dynamical systems to modified nonlinear dispersive phi-four equation, we analysis all bifurcations and phase portraits in the parametric space, the existence of solitary wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some explicit exact solution formulas are acquired for some special cases.     

Kinks and travelling wave solutions for Burgers-like equations

In this work we develop a variety of Burgers-like equations. We show that these derived equations share some of the travelling wave solutions of the Burgers equation, and differ in some other solutions.     

Exact solutions of a generalized (3 + 1)-dimensional Kadomtsev-Petviashvili equation using Lie symmetry analysis

In this paper, Lie group analysis is employed to derive some exact solutions of a generalized (3 + 1)-dimensional Kadomtsev-Petviashvili equation which describes the dynamics of solitons and nonlinear waves in plasmas and superfluids.