Solutions Of Nonlinear Klein-Gordon Equation Research Articles

Solutions of Three Dimensional Nonlinear Klein-Gordon Equations by Using Quadruple Laplace Transform

This study focuses on solving three-dimensional non-linear Klein-Gordon equations of four variables by using the quadruple Laplace transform method coupled with the iterative method. This study was designed in order to show the quadruple Laplace transform with an iterative method for solving three-dimensional nonlinear Klein-Gordon equations. The quadruple Laplace transform with the iterative method was aimed at getting analytical solutions of three-dimensional nonlinear Klein-Gordon equations. Exact solutions obtained through the iterative method have been analytically evaluated and presented in the form of a table and graph. The analytical solutions of these equations have been given in terms of convergent series with a simply calculable system, and the nonlinear terms in equations can easily be solved by the iterative method. Illustrative examples are also provided to demonstrate the applicability and efficiency of the method. The result renders the applicability and efficiency of the applied method. Finally, the quadruple Laplace transform and iterative method is an excellent method for the solution of nonlinear Klein-Gordon equations.

Analytical solutions of nonlinear Klein-Gordon equations using Multistep Modified Reduced Differential Transform Method

This paper explores the approximate analytical solution of non-linear Klein-Gordon equations (NKGE) by using multistep modified reduced differential transform method (MMRDTM). Through this proposed strategy, the non-linear term is substituted by associating Adomian polynomials obtained by utilization of a multistep approach. The NKGE solutions can be obtained with a reduced number of computed terms. In addition, the approximate solutions converge rapidly in a wide time region. Three examples are provided to illustrate the effectiveness of the proposed method to obtain solutions for the NKGE. Graphical results are shown to represent the behavior of the solution so as to demonstrate the validity and accuracy of the MMRDTM.

A numerical scheme based on differential quadrature method for numerical simulation of nonlinear Klein-Gordon equation

Purpose – The purpose of this paper is to propose a numerical scheme based on forward finite difference, quasi-linearisation process and polynomial differential quadrature method to find the numerical solutions of nonlinear Klein-Gordon equation with Dirichlet and Neumann boundary condition. Design/methodology/approach – In first step, time derivative is discretised by forward difference method. Then, quasi-linearisation process is used to tackle the non-linearity in the equation. Finally, fully discretisation by differential quadrature method (DQM) leads to a system of linear equations which is solved by Gauss-elimination method. Findings – The accuracy of the proposed method is demonstrated by several test examples. The numerical results are found to be in good agreement with the exact solutions and the numerical solutions exist in literature. The proposed scheme can be expended for multidimensional problems. Originality/value – The main advantage of the present scheme is that the scheme gives very accurate and similar results to the exact solutions by choosing less number of grid points. Secondly, the scheme gives better accuracy than (Dehghan and Shokri, 2009; Pekmen and Tezer-Sezgin, 2012) by choosing less number of grid points and big time step length. Also, the scheme can be extended for multidimensional problems.

A New Approach to Numerical Solution of Nonlinear Klein-Gordon Equation

A numerical method based on collocation points is developed to solve the nonlinear Klein-Gordon equations by using the Taylor matrix method. The method is applied to some test examples and the numerical results are compared with the exact solutions. The results reveal that the method is very effective, simple, and convenient. In addition, an error estimation of proposed method is presented.

Exact Elliptic Solution for Non-Linear Klein-Gordon Equation Via Auxiliary Equation Method

By using symbolic computation, we apply Auxiliary equation method to construct exact solutions of Non-Linear Klein-Gordon equation. We show that Auxiliary equation method provides a powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

Instability for standing waves of nonlinear Klein-Gordon equations via mountain-pass arguments

We introduce mountain-pass type arguments in the context of orbital instability for Klein-Gordon equations. Our aim is to illustrate on two examples how these arguments can be useful to simplify proofs and derive new results of orbital stability/instability. For a power-type nonlinearity, we prove that the ground states of the associated stationary equation are minimizers of the functional "action" on a wide variety of constraints. For a general nonlinearity, we extend to the dimension N=2 the classical instability result for stationary solutions of nonlinear Klein-Gordon equations proved in 1985 by Shatah in dimension N>2.

A MINIATURE SCATTERING THEORY FOR NONLINEAR KLEIN-GORDON EQUATIONS

Scattering theory is studied for small solutions of nonlinear Klein-Gordon equations in a wider Hilbert space Σ*. The nonlinearities are power-type, the order of which is less than the conformal index but more than the Strauss index. The solutions are continuous on R to Σ* and the Σ*-norm tends to zero as t →±∞. The scattering operator acts on space-time functions.

Uniform estimates for solutions of nonlinear Klein-Gordon equations

Uniform estimates in H 0 1(Ω) of global solutions to nonlinear Klein-Gordon equations of the form u tt − Δu + mu = g(u) in Ω, u = 0 in ϖΩ , where Ω is an open subset of R N , m > 0, and g satisfies some growth conditions are established.

Some Remarks on Similarity and Soliton Solutions of Nonlinear Klein-Gordon Equation

The three-dimensional nonlinear Klein-Gordon [, Higgs field and Yang-Milles] (3D-KG [, H and YM]) equation is first reduced to the 2D nonlinear Schrodinger (2D-NLS) and 2D-KG [, H and YM] equations, and secondly to the 1D-NLS and 1D-KG [, H and YM] equations by similarity transformations. It is shown that similar type soliton solutions of the 3D-KG, H and YM equations, which have singularity on a plane in ( x , y , z , t ) space, are obtained by substituting the soliton solutions of the 1D-NLS or 1D-KG (or H) equation into the similarity transformations. The soliton solutions of the YM equation are also investigated.

Scattering of solutions of nonlinear Klein-Gordon equations in three space dimensions

Soit h(u) une fonction C 1 sur R telle que h(0)=0, |h'(u)|≤c|u| P−1 , ∀u∈R. On considere l'equation de Klein-Gordon non lineaire ∂ 2 u/∂t 2 −Δu+m 2 u+h(u)=0 (x∈R 3 ,t∈R). Dans le cas 3<p<5, on montre que la theorie est vraie sous l'hypothese que les solutions sont d'energie finie

Three-dimensional multiple soliton-like solutions of non-linear Klein-Gordon equations

A method is described which enables multiple three-dimensional soliton-like solutions to be found for non-linear Klein-Gordon equations of the type Square Operator phi =F( phi ). The method is extendable to coupled equations Square Operator phi =F( phi , psi ); Square Operator psi =G( phi , psi ).

Localized solutions of non-linear Klein-gordon equations

Non-dissipative, stationary solutions for a class of non-linear Klein-Gordon equations for a scalar field have been found explicitly. Since the field is different from zero only inside a sphere of definite radius, the solutions are called quantum droplets.