Klein Gordon Schrodinger Equations Research Articles

Numerical computation of successive approximations method and variational iteration method for solving Klein-Gordon Schrödinger equation

This paper is devoted to investigating and comparing the Successive Approximations Method (SAM) and Variational Iteration Method (VIM) for solving Klein-Gordon Schrodinger (KGS) Equation. Furthermore, the approximate solutions that obtained by both methods have been represented numerically and graphically.

Error estimates for a class of energy- and Hamiltonian-preserving local discontinuous Galerkin methods for the Klein–Gordon–Schrödinger equations

The Klein–Gordon–Schrodinger (KGS) equations are classical models to describe the interaction between conservative scalar nucleons and neutral scalar mesons through Yukawa coupling. In this paper, we propose local discontinuous Galerkin (LDG) methods to solve the KGS equations. The methods involve a Crank–Nicholson time discretization for the Schrodinger equation part, a Crank–Nicholson leap frog method in time for the Klein–Gordon equation part, and local discontinuous Galerkin methods in space. Our designed numerical methods have high-order convergence rate, and energy- and Hamiltonian-preserving properties. We present the proofs of such conservation properties for both semi-discrete and fully-discrete schemes. We also establish optimal error estimates of the semi-discrete methods for the linearized KGS equations and the fully discrete methods for the KGS equations. The analysis can be extended to LDG methods for the nonlinear Klein–Gordon or Schrodinger equation, and the KGS equations in higher spatial dimensions. Several numerical tests are presented to verify some of our theoretical findings.

Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes

We establish a uniform error estimate of a finite difference method for the Klein-Gordon-Schrodinger (KGS) equations with two dimensionless parameters \begin{document}$0 and \begin{document}$0 , which are the mass ratio and inversely proportional to the speed of light, respectively. In the simultaneously nonrelativistic and massless limit regimes, i.e., \begin{document}$γ\sim\varepsilon$\end{document} and \begin{document}$\varepsilon \to 0^+$\end{document} , the KGS equations converge singularly to the Schrodinger-Yukawa (SY) equations. When \begin{document}$0 , due to the perturbation of the wave operator and/or the incompatibility of the initial data, which is described by two parameters \begin{document}$α≥0$\end{document} and \begin{document}$β≥-1$\end{document} , the solution of the KGS equations oscillates in time with \begin{document}$O(\varepsilon)$\end{document} -wavelength, which requires harsh meshing strategy for classical numerical methods. We propose a uniformly accurate method based on two key points: (ⅰ) reformulating KGS system into an asymptotic consistent formulation, and (ⅱ) applying an integral approximation of the oscillatory term. Using the energy method and the limiting equation via the SY equations with an oscillatory potential, we establish two independent error bounds at \begin{document}$O(h^2+τ^2/\varepsilon)$\end{document} and \begin{document}$O(h^2+τ^2+τ\varepsilon^{α^*}+\varepsilon^{1+α^*})$\end{document} with \begin{document}$h$\end{document} mesh size, \begin{document}$τ$\end{document} time step and \begin{document}$α^* = \min\{1, α, 1+β\}$\end{document} . This implies that the method converges uniformly and optimally with quadratic convergence rate in space and uniformly in time at \begin{document}$O(τ^{4/3})$\end{document} and \begin{document}$O(τ^{1+\frac{α^*}{2+α^*}})$\end{document} for well-prepared ( \begin{document}$α^* = 1$\end{document} ) and ill-prepared ( \begin{document}$0≤α^* ) initial data, respectively. Thus the \begin{document}$\varepsilon$\end{document} -scalability of the method is \begin{document}$τ = O(1)$\end{document} and \begin{document}$h = O(1)$\end{document} for \begin{document}$0 , which is significantly better than classical methods. Numerical results are reported to confirm our error bounds. Finally, the method is applied to study the convergence rates of KGS equations to its limiting models in the simultaneously nonrelativistic and massless limit regimes.

An improved meshless method for solving two- and three-dimensional coupled Klein–Gordon–Schrödinger equations on scattered data of general-shaped domains

In present paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to obtain the solution of two- and three-dimensional coupled Klein–Gordon–Schrodinger (KGS) equations. First, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, then we use the SMRPI approach to approximate the spatial derivatives. This approach is based on erudite combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of SMRPI. In the current work, the polyharmonic splines are used as the basis functions and to eliminate the nonlinearity, a simple predictor–corrector (P–C) scheme is performed. The aim of this paper is to show that the SMRPI procedure is suitable for the treatment of the KGS equations. The results of numerical experiments are compared with analytical solution and stability analysis is checked to confirm the accuracy and efficiency of the presented scheme.

Unconditional and optimal H 2-error estimates of two linear and conservative finite difference schemes for the Klein-Gordon-Schrödinger equation in high dimensions

The focus of this paper is on the optimal error bounds of two finite difference schemes for solving the d-dimensional (d = 2, 3) nonlinear Klein-Gordon-Schrodinger (KGS) equations. The proposed finite difference schemes not only conserve the mass and energy in the discrete level but also are efficient in practical computation because only two linear systems need to be solved at each time step. Besides the standard energy method, an induction argument as well as a ‘lifting’ technique are introduced to establish rigorously the optimal H 2-error estimates without any restrictions on the grid ratios, while the previous works either are not rigorous enough or often require certain restriction on the grid ratios. The convergence rates of the proposed schemes are proved to be at O(h 2 + τ 2) with mesh-size h and time step τ in the discrete H 2-norm. The analysis method can be directly extended to other linear finite difference schemes for solving the KGS equations in high dimensions. Numerical results are reported to confirm the theoretical analysis for the proposed finite difference schemes.

Exact solutions for the coupled Klein–Gordon–Schrödinger equations using the extended F-expansion method

The extended F-expansion method or mapping method is used to construct exact solutions for the coupled Klein-Gordon Schrodinger equations (K-G-S equations) by the aid of the symbolic computation system Mathematica. More solutions in the Jacobi elliptic function form are obtained, including the single Jacobi elliptic function solutions, combined Jacobi elliptic function solutions, rational solutions, triangular solutions, soliton solutions and combined soliton solutions.

The periodic wave solutions for the (3+1)-dimensional Klein–Gordon–Schrödinger equations

Abstract More periodic wave solutions expressed by various Jacobi elliptic functions for the Klein–Gordon–Schrodinger equations are obtained by using the extended F-expansion method. In the limit cases, the solitary wave solutions and trigonometric function solutions for the equations are also obtained.

Wave Operators for the Coupled Klein-Gordon-Schrödinger Equations in Two Space Dimensions

In this paper, we study the scattering theory for the coupled KleinGordon-Schrodinger equation (the KGS equation) with the Yukawa type interaction, which is the certain quadratic interaction, in two space dimensions. It is well-known that for the two dimensional decoupled nonlinear Schrodinger and Klein-Gordon equations with the typical nonlinearity of the form |u|p−1u, there exist wave operators if p > 2 and they do not exist otherwise. Namely, quadratic nonlinear interaction is the critical power in two space dimensions. We, however, prove the existence of wave operators to the KGS equation with that quadratic interaction for small scattered states. The proof is based on the construction of suitable second approximations of the solution to the KGS equation which imply the improved time decay estimates of the interaction terms so that the Cook-Kuroda method is applicable.

The construction of exact rational solutions with constant asymptotic values at infinity of two-dimensional NVN integrable nonlinear evolution equations via the $\overline{\partial}$-dressing method

The classes of exact rational solutions with constant asymptotic values at infinity of Nizhnik-Veselov-Novikov (NVN) equations via the -dressing method of Zakharov and Manakov are constructed. At fixed time such solutions are the transparent (exactly solvable) potentials for one-dimensional Klein-Gordon or two-dimensional stationary Schrodinger equations. Among the constructed solutions are singular and also non-singular ones.

On weakly convergent sequences in Banach function spaces and the initial-boundary value problems for non-linear Klein-Gordon-Schr�dinger equations

In the paper, we shall prove that almost everywhere convergent bounded sequence in a Banach function space X is weakly convergent if and only if X and its dual space X* have the order continuous norms. It follows that almost everywhere convergent bounded sequence in L+L (1<p1, p2<∞) is weakly convergent. On the basis of this property, we get the global existence of weak solutions for the initial–boundary value problem of non-linear Klein–Gordon–Schrodinger equations. Copyright © 2000 John Wiley & Sons, Ltd.

Spatially distributed classical Lagrangian mechanics

It is well known that the existence of two nontrivial integrals of the motion makes it possible to parametrize the motion of a Lagrangian rigid body by two variables. On the basis of this fact it is shown that certain combinations of the quantities that characterize the trajectory of such a body satisfy well-known nonlinear equations: sine—Gordon, Korteweg—de Vries, Klein-Gordon, and nonlinear Schrodinger equation.