An integrating factor for nonlinear Dirac equations

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Abstract

This paper presents an efficient integrating-factor method for solving a nonlinear Dirac equation (NLD). Starting with the simplest case of one space-variable, this method, unlike other approaches proposed in the bibliography, can be easily extended to problems with more space-variables. Our algorithm is implemented in Matlab© and the numerical experiments performed reveal its effectiveness and reliability.

Introduction

The Dirac equation is a differential equation formulated by the British physicist Paul Dirac in 1928 which provides a description of the wave function for elementary spin-12 particles such as electrons. The great achievement of this equation is its consistency with both the principles of Quantum Mechanics and the theory of Special Relativity (see, e.g., [14], [16] or [12]).

Currently, the Dirac equation for a free particle with spin-12 is written in the formiΨt=ci(k=13αkΨxk)+mc2βΨ, where Ψ=(ψ1,ψ2,ψ3,ψ4)T is the wave function; x1,x2,x3 are the spatial coordinates, t is the time; c denotes the speed of light; i is the imaginary unit; m is the mass of the electron and is Planck's constant. The standard form of the β and αk matrices in 2×2 blocks isβ=(I200I2),αk=(0σkσk0),k=1,2,3, where I2 is the 2×2 identity matrix, and σk are the Pauli matrices:σ1=(0110),σ2=(0ii0),σ3=(1001). Different types of nonlinear Dirac equations have been used in order to obtain relativistic models for other particles; we refer to [13] for a very interesting review on the historical background of this model. A general form for these equations isiΨt=ci(k=13αkΨxk)+mc2βΨF(Ψ), where the function FC2(C4,R) satisfiesF(eiϑΨ)=F(Ψ),for all ϑR. The most investigated type was proposed by Soler in [20] to describe elementary fermions. In this caseF(Ψ)=G(Ψ¯Ψ), with GC2(R,R), G(0)=0, and Ψ¯Ψ=(γ0Ψ,Ψ) the usual scalar product in C4.

In this paper, we take =c=1, and the form of the Nonlinear Dirac (NLD) equation, also known as the (3+1)-dimensional NLD equation, isΨt=A1Ψx+A2Ψy+A3Ψz+if(|Ψ1|2+|Ψ2|2|Ψ3|2|Ψ4|2)BΨ, where <x,y,z<+, 0tT, ΨC4; Aj and B are 4×4 matrices:Aj=(0σjσj0),for j=1,2,3,B=(I200I2), and f(s) is a real valued function of a real variable s. We consider the important particular casef(s)=m2λs,m,λR, where λ is the nonlinear parameter; if 0<λ1, we are dealing with a weak nonlinear problem.

This papers is organized as follows: Section 2 describes a brief review of the NLD equation, its conservation laws as well as its solitary wave solutions in the (1+1)-dimensional case and approximate solutions in the (2+1)-dimensional case. In Section 3 we compute the integrating factors corresponding to both cases and to the (3+1)-dimensional case. Section 4 reports numerical tests of the integrating factors for the (1+1)-dimensional case and (2+1)-dimensional case. The final section summarizes the results that we consider most important and we discuss what we think our next targets and inquiries should be.

Our algorithm was implemented in Matlab© and the experiments were carried out in an Intel(R) Core(TM)2 Duo CPU E6850 @ 3.00GHz.

Section snippets

Nonlinear Dirac equations

Several authors have studied the stationary solutions of (3), which are functions of the typeΨ(x,t)=eiωtΦ(x), such that Φ is a nonzero localized solution of the stationary NLD equation:0=A1Φx+A2Φy+A3Φz+if(|φ1|2+|φ2|2|φ3|2|φ4|2)BΦ. We refer to [4] for an updated review on results concerning those solutions. These results are obtained by ODE methods, variational techniques and perturbation theory; they proof the existence of solutions under different assumptions, but do not compute the

Integrating factor

The integrating factor method is based on the idea that the problem can be transformed so that the linear part of the PDE is solved exactly (see [21] and its references). In fact, it is a special case of exponential integrators, which is an active field of research and many recent publications are devoted to such numerical methods; see [11] for a review of the exponential integrators. In this section, we compute the integrating factors for the NLD equation in different dimensions.

Numerical experiments

Several authors have considered numerical methods for solving (9) in the case (5), such as finite difference methods in [2] and [1], or a split-step spectral scheme in [3]. More recently, in [8], Hong and Li consider a multi-symplectic Runge–Kutta method; more specifically, they use an implicit second-order method that preserves exactly the conservation laws, but, in practice, this is not achieved, because a fixed-point iteration method is required to solve the nonlinear algebraic systems.

Conclusions

We have presented a new numerical method to simulate the evolution in time of the nonlinear Dirac equation for one, two and three dimensions. This method uses an integrating factor to remove the linear term in the Fourier space and the resulting system of ordinary differential equations is solved by a fourth-order Runge–Kutta method. The numerical experiments have revealed that this approach is very efficient and reliable for the one space-variable case, improving in accuracy and performance

Acknowledgements

This work was supported by MEC (Spain), with the project MTM2007-62186, and by the Basque Government, with the project IT-305-07. The authors would also like to thank the anonymous referees for their careful reading of this manuscript.

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