Yongyong Caiand Yan Wang

1 School of Mathematical Sciences,Beijing Normal University,Beijing 100875,China

2 Beijing Computational Science Research Center,Beijing 100193,China

3 School of Mathematics and Statistics,Central China Normal University,Wuhan 430079,China.

Abstract.We consider the nonlinear Dirac equation(NLD)with time dependent external electro-magnetic potentials,involving a dimensionless parameterε∈(0,1]which is inversely proportional to the speed of light.In the nonrelativistic limit regimeε?1(speed of light tends to infinity),we decompose the solution into the eigenspaces associated with the‘free Dirac operator’and construct an approximation to the NLD with O(ε2)error.The NLD converges(with a phase factor)to a coupled nonlinear Schr?dinger system(NLS)with external electric potential in the nonrelativistic limit as ε→0+,and the error of the NLS approximation is first order O(ε).The constructed O(ε2)approximation is well-suited for numerical purposes.

Key words:Nonlinear Dirac equation,nonrelativistic limit,error estimates.

1 Introduction

In this paper,we consider the nonlinear Dirac equation(NLD)[3,6,14,15,23,31]in the following dimensionless form:

There have been many studies on the Dirac equations[1,8,13,16–19,21]including the well-posedness,dynamics of wave packets,etc.The purpose of this paper is to analyze the nonrelativistic limit of the nonlinear Dirac equation(1.1),whenε→0+.

For the linear case, the nonrelativistic limit has been investigated thoroughly in [7,11,20,23,25,28,30,31,33].It has been shown that the Dirac equation is a perturbation of the Schr?dinger equation asε→0+and Pauli equation is another approximation of the Dirac equation in the nonrelativistic limit regime[23,25,30,31].For the nonlinear case,Najman[30]proved that the nonrelativistic limit of the NLD(1.1)withV=Aj=0(j=1,2,3)andγ=0 is the nonlinear Schr?dinger equation for the(H2(R3))4initial data.Later,Matsuyama[27]studied the nonrelativistic limit in the weighted Sobolev space and the rapidly decreasing function space.Machiharaetal.[24]proved small global solution of NLD(1.1)and the nonrelativistic limit with(Hs(R3))4(s＞1)initial data.In[24,27,30],the nonrelativistic limit was analyzed by separating the‘spinorfield’ψinto the upper‘spinor’part(ψ1,ψ2)Tand the lower‘spinor’part(ψ3,ψ4)T.Formally,letψ:=ψ(t,x)be the solution of NLD(1.1)andφ=eitβ/ε2ψ(upper and lower‘spinors’have different phases),then the modulated functionφsatisfies(V=Aj=0(j=1,2,3)andγ=0),

Similar nonlinear Schr?dinger limit has also been found for the Klein-Gordon equation in the nonrelativistic limit[26].In[7],it has been shown that(for the linear Dirac case)the splitting of the‘spinorfield’ψas above may not be optimal in the nonrelativistic limit regime.Instead,another type splitting of the‘spinorfield’(cf.(2.10))is suggested.

This work is devoted to the study of the nonrelativistic limit of the NLD(1.1)in the general form for sufficiently smooth initial data,employing the splitting suggested in[7].We will identify the limit of the NLD(1.1)asε→0+and show the convergence rates.As expected for the linear case,the Schr?dinger limit should be a first order approximation of the NLD(1.1)in the nonrelativistic limit regime.Moreover,we shall present and analyze a second order approximation of the NLD(1.1)asε→0+(referred as seminonrelativistic limit in the later discussion),which can be viewed as an intermediate step between the NLD(1.1)and the coupled nonlinear Schr?dinger system when passing to the limitε→0+.As a common approach,Hilbert expansion can be used to construct anO(ε)correction to the nonlinear Schr?dinger system,which can yield an approximation to the NLD withO(ε2)error.However,as will be shown in the paper,our construction is more naturally and very convenient for numerical purposes(see the linear case[5]and the nonlinear case[9]).

This paper is organized as follows.In section 2,we will present the formal(semi-)nonrelativistic limit of the NLD(1.1)and the main results.The proof is shown in section 3.Finally,some conclusions and remarks are made in section 4.Throughout the paper,Crepresents a constant independent ofεand may change from line to line.We usep?qto denote that there exists a constantCindependent ofεsuch that|p|≤Cq.

2 Formal limit and the main results

For simplicity of notation,following[7],we denoteD=(D1,D2,D3)as the spatial derivative corresponding to the Fourier multiplierξ=(ξ1,ξ2,ξ3)andDj=-i?j(j=1,2,3)is the partial derivative associated toξj.Let

2.1 Formal limit

Letψε(t,x)be the solution of NLD(1.1),and define the projections ofψε(t,x)as

It can be checked that Dεis a uniformly bounded operator(w.r.t.ε)from(Hm(R3))4to(Hm-2(R3))4(m≥2).Omitting the highly oscillatory terms in(2.11)-(2.12),we obtain the following system

In the subsequent discussion,we will validate the approximation(2.14)-(2.15)and(2.17),whenε→0+.In particular,we will show that the solution of(2.14)does not exhibitε-dependent rapid oscillations whenε→0+.As a result,(2.14)is very well-suited for numerical purpose.

2.2 Main results

We make the following assumptions on the electronic potentialV(t,x)and magnetic potential A=(A1(t,x),A2(t,x),A3(t,x))T,

where 0＜T0＜∞is an arbitrary fixed time.For the initial datain(1.1),we assumeand there exists,such that

We remark here that assumption(B)is required to ensure(2.14)is anO(ε2)approximation of NLD(1.1).

Combining the linear[7,25]and the nonlinear theories[4,30],we could establish the following theorems.

Theorem 2.1.Undertheassumptions(A)and(B),thereexists0＜T1≤T0,suchthatforanyε∈(0,1],NLD(1.1)admitsauniquesolutionψε∈C1([0,T1];(Hm-1(R3))4)∩C([0,T1];(Hm(R3))4)(m≥2)withuniformestimates

whereCisindependentofε.

We have the well-posedness of(2.14)as the semi-nonrelativistic limit of the NLD(1.1).The major advantage is that the solution to the system(2.14)does not haveε-dependent highly oscillatory behavior whenε→0+.

Theorem 2.2.Undertheassumptions(A)and(B),thereexists0＜T2≤T0,suchthatfor anyε∈(0,1],(2.14)withinitialdataadmitsauniquesolutionC1([0,T2];(Hm-1(R3))4)∩C([0,T2];(Hm(R3))4)(m≥2)withuniformestimates

whereCisindependentofε.Moreover,remainintheeigenspacesassociatedwith,respectively,andifV(t,x),Aj(t,x)∈CJ-1([0,T2];Hm-J(R3)),wehave;(Hm-J(R3))4)and

Inaddition,thefollowingestimatesholdtruewhenm≥2,

whereψεisthesolutionoftheNLD(1.1)andT=min{T1,T2}.

Finally,we have the well-posedness of(2.17)as the nonrelativistic limit of the NLD(1.1).

Theorem 2.3.UnderAssumptions(A)and(B),thereexists0＜T3≤T0suchthat(2.17)with initialdataadmitsauniquesolutionφ∈C1([0,T3];(Hm-2(R3))4)∩C([0,T3];(Hm(R3))4)(m≥2).Moreover,ifm≥3,wehave

whereψεisthesolutionofNLD(1.1)andT*=min{T1,T2,T3}.

Remark 2.1.Theorems 2.1-2.3 can be easily generalized to lower dimensions.For the semi-nonrelativistic limit(2.14),one can further expandλεandw.r.t.εand omitO(ε2)terms to derive other second order approximations of the NLD(1.1).

3 The(semi-)nonrelativistic limit

We recall the following lemma regarding the projectorsfrom[7].

Lemma 3.1.([7])(i)Theprojectorsareuniformlybounded(w.r.t.ε)from(Hm(R3))4to(Hm(R3))4.

(ii)canbeexpandedas

whereR1:(Hm(R3))4→(Hm-1(R3))4andR2:(Hm(R3))4→(Hm-2(R3))4areuniformly(w.r.t.ε)boundedoperators.

3.1 Cauchy problem

In this subsection,we focus on the Cauchy problems of NLD(1.1),semi-relativistic limit system(2.14)and the nonrelativistic limit(2.17),which are involved in Theorems 2.1,2.2 and 2.3,respectively.Since the proofs are quite similar,we only prove Theorem 2.1 and sketch some of the estimates in Theorem 2.2.

ProofofTheorem2.1.By the Duhamel’s principle and using the equivalent form(2.3)of NLD(1.1),we seek a local solutionψε(t):=ψε(t,x)satisfying

It is obvious thate-itTε/ε2preserves theHmnorm and(Hm(R3))4?(L∞(R3))4which impliesF(ψ)ψis locally Lipschitz in(Hm(R3))4,i.e.,

Under assumption(A),for allt∈[0,T0],Wψis Lipschitz in(Hm(R3))4,i.e.,

whereCWdepends onand(j=1,2,3).By a standard fixed-point argument(e.g.[10]),it is easy to show that there exists a maximal timeTε∈(0,∞],such that the integral equation(3.2)admits a unique solutionψε(t,x)in the function spaceC([0,Tε];(Hm(R3))4)(m≥2).Using equation(1.1),we can verify that?tψε(t,x)∈C([0,Tε];(Hm-1(R3))4).IfTε＜+∞,we have‖ψε(t,·)‖Hm→+∞whent↗Tε.

Next,we would like to showTεhas a uniform lower boundT1＞0,i.e.,Tε≥T1(ε∈(0,1]).By the conservation of mass,we knowby assumption(B).Multiplying both sides of(1.1)by(-Δ)mψ*(we can take an approximation argument to make(-Δ)mψ*make sense inL2or simply treat it as a(H-m(R3))3)and then integrating over R3and taking the imaginary parts,using integral by parts and the Hermitian property of the“free Dirac operator”Tε,we have

whereC1depends onm,(j=1,2,3),C2depends onm.Thus,

Therefore,under assumption(B),we have a lower boundT1＞0 such thatTε≥T1(ε∈(0,1])and the estimate(2.20)holds.

ProofoftheCauchyprobleminTheorem2.2.Repeating the arguments in the proof of Theorem 2.1,using Lemma 3.1 and the fact thatλε(D)-Idis Hermitian,we can find that there existsT2＞0 such that for allε∈(0,1],(2.14)admits a unique solution(Hm-1(R3))4)∩C([0,T2];(Hm(R3))4)with uniform estimates

Using equation(2.14)and noticing the boundedness of operator Dε,we obtain

Similarly,we have the estimate for.In addition,ifV(t,x),Aj(t,x)∈CJ-1([0,T2];Hm-J(R3))have higher regularities,by differentiating(2.14)in timet,we would have(2.22).

3.2 Semi-nonrelativistic limit(2.14)

In this subsection,we want to prove Theorem 2.2.Since the Cauchy problem and the uniform estimates ofhave been considered,we only need to derive the error estimate(2.23).

Forψε,i.e.,the solution of NLD(1.1),letbe given in(2.10),i.e.,

Now,let us check the difference betweenand(the solution of(2.14)).Denote

and from the choices of initial data,we have(t=0)=0.For 0＜t≤T=min{T1,T2},we can derive the equations forby subtracting(2.14)from(2.11),

Before estimating each terms in(3.19)further,we investigate the properties of the nonlinear terms involved.

3.3 Non-relativistic limit(2.17)

Here,we would like to show the nonrelativistic limit of NLD(1.1)is(2.17).As mentioned,it suffices to prove that the coupled system(2.14)-(2.15)converges to(2.16)asε→0+.Now,we shall assumem≥3.

4 Conclusion

The nonlinear Dirac equation with external electro-magnetic potentials was considered.There is a dimensionless parameterεinversely proportional to the speed of light in the NLD.We investigated the NLD in the nonrelativistic limit regime,i.e.,when 0＜ε?1.By projecting the solution into the eigenspaces of the“free Dirac operator”,we obtained the coupled nonlinear Schr?dinger system as the nonrelativistic limit and a second order approximation as the semi-nonrelativistic limit,whenε→0+.The analysis provided the detailed structure of the NLD in the nonrelativistic limit.In particular,the seminonrelativistic limit,which is anO(ε2)approximation to the NLD,is well-suited for numerical implementations,since the differential operators can be easily calculated in Fourier space.