Lie group symmetries and Riemann function of Klein–Gordon–Fock equation with central symmetry

In the present paper Lie symmetry group method is applied to find new exact invariant solutions for Klein–Gordon–Fock equation with central symmetry. The found invariant solutions are important for testing finite-difference computational schemes of various boundary value problems of Klein–Gordon–Fock equation with central symmetry. The classical admitted symmetries of the equation are found. The infinitesimal symmetries of the equation are used to find the Riemann function constructively.     

Application of homotopy-perturbation method to Klein–Gordon and sine-Gordon equations

In this paper, the homotopy-perturbation method (HPM) is employed to obtain approximate analytical solutions of the Klein–Gordon and sine-Gordon equations. An efficient way of choosing the initial approximation is presented. Comparisons with the exact solutions, the solutions obtained by the Adomian decomposition method (ADM) and the variational iteration method (VIM) show the potential of HPM in solving nonlinear partial differential equations.     

Solution of non-linear Klein–Gordon equation with a quadratic non-linear term by Adomian decomposition method

Non-linear PDEs are systematically solved by the decomposition method of Adomian for general boundary conditions described by boundary operator equations. In the present case the solution of the non-linear Klein–Gordon equation has been considered as an illustration of the decomposition method of Adomian.     

Klein–Gordon equations with homogeneous time-dependent electric fields

We consider a system associated to Klein–Gordon equations with homogeneous time-dependent electric fields. The upper and lower boundaries of a time-evolution propagator for this system were proven by Veseli? (J Oper Theory 25:319–330, 1991) for electric fields that are independent of time. We extend this result to time-dependent electric fields.     

Conservative difference methods for the Klein–Gordon–Zakharov equations

Firstly an implicit conservative finite difference scheme is presented for the initial-boundary problem of the one space dimensional Klein–Gordon–Zakharov (KGZ) equations. The existence of the difference solution is proved by Leray–Schauder fixed point theorem. It is proved by the discrete energy method that the scheme is uniquely solvable, unconditionally stable and second order convergent for U   in _l∞$l_{\infty }$ norm, and for N   in _l2$l_2$ norm on the basis of the priori estimates. Then an explicit difference scheme is proposed for the KGZ equations, on the basis of priori estimates and two important inequalities about norms, convergence of the difference solutions is proved. Because it is explicit and not coupled it can be computed by a parallel method. Numerical experiments with the two schemes are done for several test cases. Computational results demonstrate that the two schemes are accurate and efficient.     

Energy bounds for Klein–Gordon equations with time-dependent potential

The goal of the present paper is to explain the interplay between a decreasing and an oscillating part of a time-dependent coefficient in the mass of a linear Klein–Gordon type model. Lower bounds of suitable energies for t → ∞ are derived by using Floquet’s theory for Hill’s equation with two parameters coupled with an instability argument. The results exclude the property of generalized energy conservation for the class of models under consideration.     

Soliton perturbation theory for phi-four model and nonlinear Klein–Gordon equations

This paper obtains the adiabatic variation of the soliton velocity, in presence of perturbation terms, of the phi-four model and the nonlinear Klein–Gordon equations. There are three types of models of the nonlinear Klein–Gordon equation, with power law nonlinearity, that are studied in this paper. The soliton perturbation theory is utilized to carry out this investigation.     

Small generalized breathers with exponentially small tails for Klein–Gordon equations

We consider a class of nonlinear Klein–Gordon equation _utt=_uxx−u+f(u)$u_{tt}=u_{xx}-u+f(u)$ and show that generically there exist small breathers with exponentially small tails.     

Solitary waves for nonlinear Klein–Gordon equations coupled with Born–Infeld theory

We consider the nonlinear Klein–Gordon equations coupled with the Born–Infeld theory under the electrostatic solitary wave ansatz. The existence of the least-action solitary waves is proved in both bounded smooth domain case and R³${\mathbb{R}}^3$ case. In particular, for bounded smooth domain case, we study the asymptotic behaviors and profiles of the positive least-action solitary waves with respect to the frequency parameter ω. We show that when κ and ω   are suitably large, the least-action solitary waves admit only one local maximum point. When ω→∞$\omega \to \infty$, the point-condensation phenomenon occurs if we consider the normalized least-action solitary waves.