On the time-splitting Fourier spectral method for the coupled Klein-Gordon-Schrödinger equations in the quantum field theory

Abstract

Purpose

The purpose of this paper is to consider the time-splitting Fourier spectral (TSFS) method to solve the fractional coupled Klein–Gordon–Schrödinger (K-G-S) equations. A time-splitting spectral approach is applied for discretizing the Schrödinger-like equation and along with that, a pseudospectral discretization has been accurately utilized for the temporal derivatives in the Klein–Gordon-like equation. Furthermore, the time-splitting scheme is proved to be unconditionally stable. Numerical experiments guarantee high accuracy of the TSFS scheme for the K-G-S equations. Here, the derivative of fractional order is taken in the Riesz sense.

Design/methodology/approach

The focus of this paper is to study the Riesz fractional coupled K-G-S equations using the TSFS method. This method is dependent on evaluating the solution to the given problem in small steps, and treating the nonlinear and linear steps separately. The nonlinear step is made in the time domain, while the linear step is made in the frequency domain, which necessitates the use of Fourier transform back and forth. It is a very effective, powerful and efficient method to solve the nonlinear differential equations, as in previous works (Bao et al., 2002; Bao and Yang, 2007; Muslu and Erbay, 2003; Borluk et al., 2007), the initial and boundary-value problem is decomposed into linear and nonlinear subproblems. Summarizing the technique of the TSFS method, it can be stated that first the Schrödinger-like equation is solved in two splitting steps. Then, the Klein–Gordon-like equation is solved by discretizing the spatial derivatives by means of the pseudospectral method.

Findings

The utilized method is found to be very efficient and accurate. Moreover, the time-splitting spectral scheme is found to be unconditionally stable. By means of thorough study, it is found that the spectral method is time-reversible, is gauge-invariant and also conserves the total charge. Moreover, the results have been graphically presented to exhibit the accuracy of the proposed methods. Apart from that, the numerical solutions have been also compared with the exact solutions. Numerical experiments establish that the proposed technique manifests high accuracy and efficiency.

Originality/value

To the authors’ best knowledge, the Riesz fractional coupled K-G-S equations have been for the first time solved by using the TSFS method.

Citation

Saha Ray, S. (2017), "On the time-splitting Fourier spectral method for the coupled Klein-Gordon-Schrödinger equations in the quantum field theory", Engineering Computations, Vol. 34 No. 7, pp. 2344-2357. https://doi.org/10.1108/EC-04-2017-0132