Conservative compact finite difference scheme for the N‐coupled nonlinear Klein–Gordon equations

In this article, a compact finite difference method is developed for the periodic initial value problem of the N‐coupled nonlinear Klein–Gordon equations. The present scheme is proved to preserve the total energy in the discrete sense. Due to the difficulty in obtaining the priori estimate from the discrete energy conservation law, the cut‐off function technique is employed to prove the convergence, which shows the new scheme possesses second order accuracy in time and fourth order accuracy in space, respectively. Additionally, several numerical results are reported to confirm our theoretical analysis. Lastly, we apply the reliable method to simulate and study the collisions of solitary waves numerically.     

A coupled method of Laplace transform and Legendre wavelets for nonlinear Klein–Gordon equations

Klein–Gordon equation models many phenomena in both physics and applied mathematics. In this paper, a coupled method of Laplace transform and Legendre wavelets, named (LLWM), is presented for the approximate solutions of nonlinear Klein–Gordon equations. By employing Laplace operator and Legendre wavelets operational matrices, the Klein–Gordon equation is converted into an algebraic system. Hence, the unknown Legendre wavelets coefficients are calculated in the form of series whose components are computed by applying a recursive relation. Block pulse functions are used to calculate the Legendre wavelets coefficient vectors of nonlinear terms. The convergence analysis of the LLWM is discussed. The results show that LLWM is very effective and easy to implement. Copyright © 2013 John Wiley & Sons, Ltd.     

Wave operator for the system of the Dirac–Klein–Gordon equations

We prove the existence of the wave operator for the system of the massive Dirac–Klein–Gordon equations in three space dimensions x∈ R ³ where the masses m, M>0. We prove that for the small final data , (?, ?)∈ H ^{2 + µ, 1} × H ^{1 + µ, 1}, with and , there exists a unique global solution for system (1) with the final state conditions Copyright © 2010 John Wiley & Sons, Ltd.     

Uniform decay and blow‐up of solutions for coupled nonlinear Klein–Gordon equations with nonlinear damping terms

In this work, we consider coupled nonlinear Klein–Gordon equations with nonlinear damping terms, in a bounded domain. The decay estimates of the solution are established by using Nakao's inequality. We also prove the blow up of the solution in finite time with negative initial energy. Copyright © 2013 John Wiley & Sons, Ltd.     

From the Klein–Gordon–Zakharov system to the Klein–Gordon equation

In a singular limit, the Klein–Gordon (KG) equation can be derived from the Klein–Gordon–Zakharov (KGZ) system. We point out that for the original system posed on a d‐dimensional torus, the solutions of the KG equation do not approximate the solutions of the KGZ system. The KG system has to be modified to make correct predictions about the dynamics of the KGZ system. We explain that this modification is not necessary for the approximation result for the whole space with d≥3. Copyright © 2016 John Wiley & Sons, Ltd.     

Asymptotic profile of solutions for strongly damped Klein‐Gordon equations

We consider the Cauchy problem in R ⁿ for strongly damped Klein‐Gordon equations. We derive asymptotic profiles of solutions with weighted L^1,1( R ⁿ) initial data by a simple method introduced by the second author. Furthermore, from the obtained asymptotic profile, we get the optimal decay order of the L²‐norm of solutions. The obtained results show that the wave effect will be relatively weak because of the mass term, especially in the low‐dimensional case (n = 1,2) as compared with the strongly damped wave equations without mass term (m = 0), so the most interesting topic in this paper is the n = 1,2 cases to compare the difference.     

On the numerical solution of the Klein‐Gordon equation

A predictor–corrector (P–C) scheme based on the use of rational approximants of second‐order to the matrix‐exponential term in a three‐time level reccurence relation is applied to the nonlinear Klein‐Gordon equation. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. Both the predictor and the corrector scheme are analyzed for local truncation error and stability. The proposed method is applied to problems possessing periodic, kinks and single, double‐soliton waves. The accuracy as well as the long time behavior of the proposed scheme is discussed. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

An accurate numerical method for solving the linear fractional Klein–Gordon equation

In this article, an implementation of an efficient numerical method for solving the linear fractional Klein–Gordon equation (LFKGE) is introduced. The fractional derivative is described in the Caputo sense. The method is based upon a combination between the properties of the Chebyshev approximations and finite difference method (FDM). The proposed method reduces LFKGE to a system of ODEs, which is solved using FDM. Special attention is given to study the convergence analysis and deduce an error upper bound of the proposed method. Numerical example is given to show the validity and the accuracy of the proposed algorithm. Copyright © 2013 John Wiley & Sons, Ltd.     

BEM‐FEM coupling for the 1D Klein–Gordon equation

A transmission (bidomain) problem for the one‐dimensional Klein–Gordon equation on an unbounded interval is numerically solved by a boundary element method‐finite element method (BEM‐FEM) coupling procedure. We prove stability and convergence of the proposed method by means of energy arguments. Several numerical results are presented, confirming theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 2042–2082, 2014     

Exact solutions of multi-component nonlinear Schrödinger and Klein–Gordon equations

Applying the sec_q−tanh_q-method [Phys. Lett. A 298 (2002) 253], we find a class of exact solution of multi-component nonlinear Schrödinger and Klein–Gordon equations and generalize the correspond results in [Phys. Lett. A 298 (2002) 253] and [J. Phys. A: Math. Gen. 34 (2001) 4281].     

Stability analysis and a numerical scheme for fractional Klein‐Gordon equations

Fractional order nonlinear Klein‐Gordon equations (KGEs) have been widely studied in the fields like; nonlinear optics, solid state physics, and quantum field theory. In this article, with help of the Sumudu decomposition method (SDM), a numerical scheme is developed for the solution of fractional order nonlinear KGEs involving the Caputo's fractional derivative. The coupled method provides us very efficient numerical scheme in terms of convergent series. The iterative scheme is applied to illustrative examples for the demonstration and applications.     

Periodic travelling waves in convex Klein–Gordon chains

We study Klein–Gordon chains with attractive nearest neighbour forces and convex on-site potential, and show that there exists a two-parameter family of periodic travelling waves (wave trains) with unimodal and even profile functions. Our existence proof is based on a saddle-point problem with constraints and exploits the invariance properties of an improvement operator. Finally, we discuss the numerical computation of wave trains.     

Orbital stability of solitary waves of the coupled Klein–Gordon–Zakharov equations

This paper investigates the orbital stability of solitary waves for the coupled Klein–Gordon–Zakharov (KGZ) equations where α ≠ 0. Firstly, we rewrite the coupled KGZ equations to obtain its Hamiltonian form. And then, we present a pair of sech‐type solutions of the coupled KGZ equations. Because the abstract orbital stability theory presented by Grillakis, Shatah, and Strauss (1987) cannot be applied directly, we can extend the abstract stability theory and use the detailed spectral analysis to obtain the stability of the solitary waves for the coupled KGZ equations. In our work, α = 1,β = 0 are advisable. Hence, we can also obtain the orbital stability of solitary waves for the classical KGZ equations which was studied by Chen. Copyright © 2016 John Wiley & Sons, Ltd.     

On a factorization of the Schrödinger and Klein–Gordon operators

A general scheme for factorizing second‐order time‐dependent operators of mathematical physics is given, which allows a reduction of corresponding second‐order equations to biquaternionic equations of first order. Examples of application of the proposed scheme are presented for both constant and variable coefficients. Copyright © 2008 John Wiley & Sons, Ltd.     

The Klein–Gordon Equation in a Domain with Time‐Dependent Boundary

We solve a Dirichlet boundary value problem for the Klein–Gordon equation posed in a time‐dependent domain. Our approach is based on a general transform method for solving boundary value problems for linear and integrable nonlinear PDE in two variables. Our results consist of the inversion formula for a generalized Fourier transform, and of the application of this generalized transform to the solution of the boundary value problem.     

On the complete group classification of the one‐dimensional nonlinear Klein–Gordon equation with a delay

This research gives a complete Lie group classification of the one‐dimensional nonlinear delay Klein–Gordon equation. First, the determining equations are derived and their complete solutions are found. Then the complete group classification and representations of all invariant solutions are obtained. Copyright © 2015 John Wiley & Sons, Ltd.     

A moving Kriging‐based MLPG method for nonlinear Klein–Gordon equation

In this paper, the meshless local Petrov–Galerkin approximation is proposed to solve the 2‐D nonlinear Klein–Gordon equation. We used the moving Kriging interpolation instead of the MLS approximation to construct the meshless local Petrov–Galerkin shape functions. These shape functions possess the Kronecker delta function property. The Heaviside step function is used as a test function over the local sub‐domains. Here, no mesh is needed neither for integration of the local weak form nor for construction of the shape functions. So the present method is a truly meshless method. We employ a time‐stepping method to deal with the time derivative and a predictor–corrector scheme to eliminate the nonlinearity. Several examples are performed and compared with analytical solutions and with the results reported in the extant literature to illustrate the accuracy and efficiency of the presented method. Copyright © 2016 John Wiley & Sons, Ltd.     

Stability and instability of periodic standing wave solutions for some Klein–Gordon equations

In the present paper we show some results concerning the orbital stability of dnoidal standing wave solutions and orbital instability of cnoidal standing wave solutions to the following Klein–Gordon equation:

u_tt−u_xx+u−|u|²u=0.