Klein-Gordon方程的定性分析及精确解

本文运用动力系统的定性理论,得到了Klein-Gordon方程的孤立波分支图,并求得了与定性分析结果一致的孤波解.     

Numerical solution of the nonlinear Klein-Gordon equation

A numerical method is developed to solve the nonlinear one-dimensional Klein-Gordon equation by using the cubic B-spline collocation method on the uniform mesh points. We solve the problem for both Dirichlet and Neumann boundary conditions. The convergence and stability of the method are proved. The method is applied on some test examples, and the numerical results have been compared with the exact solutions. The L₂, L_∞ and Root-Mean-Square errors (RMS) in the solutions show the efficiency of the method computationally.     

Exact travelling wave solutions of the dissipative coupled Korteweg-de Vries equation

This paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions of the dissipative coupled Korteweg-de Vries equation. The possible kink profile solitary wave solutions and approximate damped oscillatory solutions of the equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful.     

A spline collocation approach for the numerical solution of a generalized nonlinear Klein-Gordon equation

In this paper, a finite element collocation approach using cubic B-splines is employed for the numerical solution of a generalized form of the nonlinear Klein-Gordon equation. The efficiency of the method is tested on a number of examples that represent special cases of the extended equation including the sine-Gordon equation. The numerical results are compared with existing numerical and analytic solutions and the outcomes confirm that the scheme yields accurate and reliable results even when few nodes are used at the time levels.     

Exact solitary wave solutions of the Kadomtsov-Petviashvili-Benjamin-Bona-Mahony equation

Bifurcation method of dynamical systems is employed to investigate bifurcation of solitary waves in the nonlinear dispersive Kadomtsov-Petviashvili-Benjamin-Bona-Mahony equation. Numbers of solitary waves are given for each parameter condition. Under some parameter conditions, exact solitary wave solutions are obtained.     

Exact traveling wave solutions of the Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation

Bifurcation method of dynamical systems is employed to investigate traveling wave solutions in the (2 + 1)-dimensional Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation. Under some parameter conditions, exact solitary wave solutions and kink wave solutions are obtained.     

浅水波方程和Klein-Gordon方程新的精确行波解

采用了一种新的方法来求解浅水波方程和Klein-Gordon的行波解.在该方法下,Klein-Gordon方程和浅水波方程都得到了其精确的周期孤立波解,从而该方法的有效性得到了验证.     

Singular soliton solution and bifurcation analysis of Klein-Gordon equation with power law nonlinearity

In this paper, the Klein-Gordon equation (KGE) with power law nonlinearity will be considered. The bifurcation analysis as well as the ansatz method of integration will be applied to extract soliton and other wave solutions. In particular, the second approach to integration will lead to a singular soliton solution. However, the bifurcation analysis will reveal several other solutions that are of prime importance in relativistic quantum mechanics where this equation appears.     

A new method to solve the damped nonlinear Klein-Gordon equation

This paper discusses a damped nonlinear Klein-Gordon equation in the reproducing kernel space and provides a new method for solving the damped nonlinear Klein-Gordon equation based on the reproducing kernel space.Two numerical examples are given for illustrating the feasibility and accuracy of the method.     

Exact traveling wave solutions for a modified Camassa-Holm equation

In this paper, the bifurcation method of planar dynamical systems is utilized to investigate a modified Camassa-Holm equation. After dividing the parametric space, some explicit parametric conditions are derived for the existence of traveling wave solutions. Several exact traveling solutions are also 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.     

Global solutions and finite time blow up for damped Klein-Gordon equation

We study the Cauchy problem of strongly damped Klein-Gordon equation. Global existence and asymptotic behavior of solutions with initial data in the potential well are derived. Moreover, not only does finite time blow up with initial data in the unstable set is proved, but also blow up results with arbitrary positive initial energy are obtained.     

Solitary wave solutions to the generalized coupled mKdV equation with multi-component

Using the Darboux matrix method, the multi-solitary wave solutions of the generalized coupled mKdV equation with multi-component are obtained. The obtained solution formulas provide us with a comprehensive approach to construct exact solutions for the generalized coupled mKdV equation by some basic solutions of the Boiti and Tu spectral problem.     

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.     

Exact three-wave solutions for the KP equation

A new type of three-wave solution, periodic two-solitary-wave solutions, for (1 + 2)D Kadomtsev-Petviashvili (KP) equation is obtained using the extended three-soliton method and with the help of Maple.     

Stability of the standing waves for a class of coupled nonlinear Klein-Gordon equations

This paper deals with the standing waves for a class of coupled nonlinear Klein-Gordon equations with space dimension N ≥ 3, 0 〈 p, q 〈 2/N-2 and p ＋ q 〈 4/N. By using the variational calculus and scaling argument, we establish the existence of standing waves with ground state, discuss the behavior of standing waves as a function of the frequency ω and give the sufficient conditions of the stability of the standing waves with the least energy for the equations under study.     

New explicit solutions for the Klein-Gordon equation with cubic nonlinearity

In this paper, we investigate Klein-Gordon equation with cubic nonlinearity. All explicit expressions of the bounded travelling wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain bell-shaped solitary wave solutions, kink-shaped solitary wave solutions and Jacobi elliptic function periodic solutions. Moreover, we point out the region which these periodic wave solutions lie in. We present the relation between the bounded travelling wave solution and the energy level h. We find that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it is shown that the exact periodic solutions evolute into solitary wave solution.     

Exact solutions of Kolmogorov-Petrovskii-Piskunov equation using the modified simple equation method

The modified simple equation method is employed to find the exact solutions of the nonlinear Kolmogorov-Petrovskii-Piskunov （KPP） equation. When certain parameters of the equations are chosen to be special values, the solitary wave solutions are derived from the exact solutions. It is shown that the modified simple equation method provides an effective and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.     

New solitons and kink solutions for the Gardner equation

The Gardner equation, also called combined KdV–mKdV equation, is studied. New hyperbolic ansatze are proposed to derive solitons solutions. The tanh method is used as well to obtain kink solutions.