Existence and Uniqueness of the Global Smooth Solution to the Periodic Boundary Value Problem of Fractional Nonlinear Schr¨odinger System

WEI Gongmingand DONG Jing

College of Science,University of Shanghai for Science and Technology, Shanghai 200093,China.

Existence and Uniqueness of the Global Smooth Solution to the Periodic Boundary Value Problem of Fractional Nonlinear Schr¨odinger System

WEI Gongming?and DONG Jing

College of Science,University of Shanghai for Science and Technology, Shanghai 200093,China.

.In this paper,we study a class of coupled fractional nonlinear Schr¨odinger system with periodic boundary condition.Using Galerkin method,the existence of global smooth solution is obtained.We also prove the uniqueness of the solution.

Nonlinear Schr¨odinger system;global smooth solution;Gagliardo-Nirenberg inequality.

1 Introduction

The Schr¨odinger equation is the fundamental system of physics for describing nonrelativistic quantum mechanical behavior.It is well known that Feynman and Hibbs[1] used path integrals over Brownian paths to derive the standard(non-fractional)nonlinear Schr¨odinger equation.Recently,Laskin[2,3]showed that the path integral over the L′evy-like quantum mechanical paths allows to develop the generalization of the quantum mechanics.Namely,if the path integral over Brownian trajectories leads to the well known Schr¨odinger equation,then the path integral over L′evy trajectories leads to the fractional Schr¨odinger equation.The fractional Schr¨odinger equation includes the space derivative of orderαinstead of the second(α=1)order space derivative in the standard Schr¨odinger equation.Laskin[4]showed the Hermiticity of the fractional Hamilton operatorand established the parity conservation law.Guo and Xu[5]studied some physical applications of the fractional Schr¨odinger equation.

Nonlinear equations aroused the interest of a large number of mathematicians.In paper[6],Guo and Wang,by using Galerkin method,fixed point principle,and sequential approximation method,proved the existence of periodic solutions for the nonlinear systems of Schr¨odinger equations with Dirichlet boundary conditions and periodic condition in time.Guo[7,8]showed the existence and nonexistence of a global solution for one class of the strong nonlinear Schr¨odinger equations of fourth order,and studied the initial value problem of integral nonlinear Schr¨odinger system.In[9]Guo considered a class of high order multidimensional nonlinear systems of Schr¨odinger equations for the initial boundary value and the periodic initial value problem.

In this paper,we consider the following fractional nonlinear Schr¨odinger system

with the following initial condition and periodic boundary condition

Whenα=1,(1.1)becomes the standard nonlinear Schro¨dinger system which has been studied by many works,e.g.[6].The existence and uniqueness of the weak solution to the initial-boundary value problem can be found in[10].The existence of the global smooth solution to the initial-boundary value problem was studied in[11,12].However, there is few mathematical analysis for the fractional nonlinear Schro¨dinger system(1.1).

In this paper,we prove the existence of the global smooth solution(as smooth in theHα)to the period boundary value problem of fractional nonlinear Schro¨dinger system (1.1)by using the energy method.Theorems 2.1 and 4.1 are the new fundamental results of the fractional nonlinear Schro¨dinger system.We obtain respectively the global existence of smooth solution to the period boundary value problem of the fractional nonlinear Schro¨dinger system in Theorems 2.1 and 4.1.

2 Preliminary and main results

Letube a periodic function,we can expressuby a Fourier series:

LetHαbe a complete space of the setAwith the norm:

ThenHαis a Banach space.It is easy to show thatHαis also a Hilbert space with the inner product

LetXdenote a Banach space with normWe need the following definitions.

Definition 2.1.The space LP(0,T;X)consists of all measurable functions f:[0,T]→X with

Definition 2.2.The spaceconsists of all continuous functionswith

Lemma 2.3.[13,p.709](Gronwall’s inequality in differential form)

(i)Let η(·)be a nonnegative,absolutely continuous function on[0,T],which satisfies for a.e. t the differential inequality,

where φ(t)and ?(t)are nonnegative,summable functions on[0,T].Then

(ii)In particular,if

Lemma 2.4.[14,p.260](Gagliardo-Nirenberg inequality)

and the optimal constant S has the explicit form

The main result in this paper is as follows.

Theorem 2.5.LetIf ρ is an even,suppose that ρ＞0whenβ＞0and0＜ρ＜4α/n when β＜0.If ρ is odd,suppose that ρ＞2[α]+1when β＞0and2[α]+1＜ρ＜4α/n when β＜0.Then for all u0,ν0∈H4α(?),there exists a global smooth solution u(x,t)and ν(x,t)of the problem(1.1)-(1.3),such that:

Theorem 2.5 generalizes the result of the nonlinear Schr¨odinger equations in[3].

3 A priori estimates

We will prove the main result by Galerkin method,i.e.first construct approximate solutions and then prove that the sequence of approximate solutions converges to a solution of the original problem.In order to prove the convergence of these approximate solutions,we need to obtain a priori estimates of these approximate solutions.Therefore, we first prove a priori estimates of the problem(1.1)-(1.3),which are the same as the approximate solutions satisfy.

Lemma 3.1.Suppose that α＞0and q＞0,γ,ζ＞0and γ,ζ∈R,u(x,t),ν(x,t)solves the problem(1.1)-(1.3).Then

Proof.Multiply(1.1)byintegrate with respecttox,and then take the imaginary part, we have

Multiply the first equation byγ,the second equation byζ,and do addition,we have

Throughout this paper,Twill denote an arbitrary positive constant andcdenotes a positive constant which depends only on initial data andT.

Lemma 3.2.Let α＞0.Suppose that ρ＞0when β＞0and0＜ρ＜4α/n when β＜0,u=u(x,t),ν=ν(x,t)solves the problem(1.1)-(1.3).Then

Here c depends on the Hαand Lρ+2norm of u0or ν0.

Proof.Multiply(1.1)byintegrate with respect tox,then we obtain

Multiply the first equation byγ,the first equation byζ,do addition,and take the real part,then we have

By(3.1)and the above equality,we get

In the following,we consider the caseβ＜0.Letθ=nρ/2α(ρ+2)＜1.Then

By Gagliardo-Nirenberg inequality(derived from in[12,Lemma 3.2],[14]and[8,Lemma 2],we have

Sinceρ＜4α/n,nρ/2α＜2,from the above inequalities,we get

By(3.3)and(3.4),whenβ＜0,we get

Note thatγ＞0,ζ＞0.Hence

Lemma 3.3.Let α＞n/2.Suppose that ρ satisfy the condition of Lemma3.2,u=u(x,t),ν=ν(x,t)solves the problem(1.1)-(1.3).Then

for constant c＞0and c should not depend on u,ν.

Proof.Differentiate(1.1)with respect to t,multiply byand integrate with respect tox,then we obtain

Multiply the first equation byγ,the first equation byζ,do addition,and take the imaginary part,then we get

But by derivation process of Lemma 3.3 in[12],

Then by(3.6)and(3.7),we obtain

Integrating the above equality from 0 tot,we have

By applying Eq.(1.1),Definition 2.1.1 in[15]and derivation process of Lemma 3.3 in [12],we have

Again by derivation process of Lemma 3.3 in[12]and note thatα＞n/2,we have

Then from(3.8),we get

By Gronwall inequality,we deduce that

By(1.1)and the above inequality,we obtain

and hence

Lemma 3.4.Let α＞n/2.Suppose that ρ satisfy the condition of Lemma3.2,if ρ is not an even, ρ＞[α],when β＞0,and[α]＜ρ＜4α/n when β＜0.Assume that u=(x,t),ν=(x,t)solves the problem(1.1)-(1.3).Then

for constant c should depend on ρ,β,γ,ζ.

Proof.Differentiate(1.1)with respect to t,multiply byˉutt,ˉνtt,and then integrate with respect toxto obtain

Then multiply the first equation byγ,the second equation byζ,take the real part of the above two equalities,and we get

From the derivation process of Lemma 3.4 in[12],we get

From(3.10)and(3.11),we have

Add those two equations,we have

The above equality implies that

Here the constantcshould not depend onu,ν,ut,νt.

Letθ=n/6α＜1/3.Then

Thus,by Gagliardo-Nirenberg inequality[12,14]and(3.5),we have

From(3.12)and(3.13),we obtain

In fact,by(1.1),Definition 2.1.1 in[15,p.37],we have

where we use the condition thatρ＞[α]ifρis not an even.

By(3.13),we have

From(3.14),(3.15)and Gronwall’s inequality,we deduce that

Lemma 3.5.Let α＞n/2.Suppose that ρ satisfies the conditions of Lemma3.2if ρ is an even. If ρ is not an even,suppose that ρ＞2[α]+1when β＞0and2[α]+1＜ρ＜4α/n when β＜0.

Assume that u=u(x,t),ν=ν(x,t)solves the problem(1.1)-(1.3).Then

Here the constant c also depends on ρ,β,γ,ζ.

Proof.Differentiate(1.1)with respect to t two times,multiply byand then integrate with respect toxto obtain that

Multiply the above two equations by theζandγ,respectively,and do addition,then we have

By taking the imaginary part of the above equality,we get

For the first term of the right hand side of(3.18),we have

Similarly,for the second term and the third term to the right hand side of(3.18),we have

For the last term to the right hand side of(3.18),we get

By(3.17)-(3.21),we conclude that

By(3.5),(3.9)and Gagliardo-Nirenberg inequality[12,14],we have

While,by(1.1)and(3.5),we have

Ifα≥max{n/2,1},by(3.25)in[12],we have

where we use the conditionρ＞2[α]+1 ifρis not an even.Ifn=1 and 1/2＜α＜1,

From(3.24)-(3.26),we have

From(3.22)and(3.27),we have

Therefore,Gronwall’s inequality implies that

By(1.1),[12,p.473]and the above inequality,we have

From(3.28)and the above inequality,we have

Lemma 3.6.Suppose that α and ρ satisfy the conditions of Lemma3.5,u=u(x,t),ν=ν(x,t)solves the problem(1.1)-(1.3).Then,

Proof.Whenα≥max{n/2,1},by(1.1),(3.5)and(3.29),we obtain

Whenn=1 and 1/2＜α＜1,by(1.1)and(3.16),we have

By Gagliardo-Nirenberg inequality[12,14]and(3.1),we have

By Gagliardo-Nirenberg inequality[12,14]and(3.5),we have

By(3.32)-(3.34),we conclude that

Whenn=1 and 1/2＜α＜1,from(3.31)and the above inequality,we complete the proof.

Finally,whenα≥max{n/2,1},by[12],

Whenn=1 and 1/2＜α＜1,by[12],we have

Then,by[12],we conclude that

In the same way,we can get that

4 Proof of the main result

Before proving Theorem 2.5,we first prove the existence of the weak solution to the problem(1.1)-(1.3)by using Faedo-Galerkin’s method.We need the following lemmas.

Lemma 4.1.[12,p.474]Let B0,B and B1be three Banach spaces.Assume that B0?B?B1and Bi,i=0,1,are reflective.Suppose also that B0is compactly embedded in B.Let

where T is finite and1＜pi＜∞,i=0,1,W is equipped with the norm

Then W is compactly embedded in Lp0(0,T;B).

Lemma4.2.[12,p.475]Supposethat Q isaboundeddomain inFurthermore,suppose that

Lemma 4.3.[12,p.475]X is a Banach space.Suppose that(after possibly being redefined on a set of measure zero).

In the following,we prove the existence of weak solution to the problem(1.1)-(1.3).

Theorem 4.4.Let α＞0.When β＞0,suppose that ρ＞0if α≥n/2and0＜ρ＜4α/(n?2α)if α＜n/2.When β＜0,suppose that0＜ρ＜4α/n,u0,ν0∈Hα(?).Then there exists a function u=u(x,t),ν=ν(x,t)satisfying(1.1)-(1.3),such that

Proof.We prove Theorem 4.4 by three steps.

Step 1.Construction of approximate solutions by Faedo-Galerkin’s method

Fix now a positive integer m,we will look for a functionum=um(t),νm=νm(t)of the form

Then(4.3)becomes the system of nonlinear ODE subject to the initial condition(4.4).According to standard existence theory for nonlinear ordinary differential equations,there exists a unique solution of(4.3)and(4.4)for a.e.0≤t≤tm.By a priori estimates we obtain thattm=Tby Theorem 2 of[6].

Step 2.A priori estimates

As in the proof of Lemmas 3.1,3.2 and from[12,p.475],we have

For any?∈Hα(?),we have

By[12,p.475],we have

By Sobolev imbedding theorem and derivation process in[12,p475],we have

By(4.6)and(4.7),we have

Step 3.Passaging to the limit

By applying(4.5),(4.8)and[8],we deduce that there exists a subsequenceuμ,vμ,uμ+vμfromum,νm,um+νm,such that:

By(4.5),we have

By(4.8),we have

We equipWwith the norm

SinceHα(?)is compactly embedded inL2(?),Wis compactly embedded inL20,T;H2(?) by Lemma 4.1.By(4.11)and(4.12),um∈W,νm∈W.Then,there exists the subsequenceuμ,νμ,uμ+νμ(not relabelled)which satis fies

By(4.5),(4.13)and Lemma 4.2,we have

Fixingj,by(4.3),we get

By applying(4.9),(4.10)and(4.14),we deduce that there exists a subsequenceuμ,νμfromum,νm,such that

From(4.15),we have

The above equality holds for any fixedj.By the density of the basiswj(j∈Zn),we have

Henceu,νsatisfies(1.1)and(4.1).

By(4.5),(4.8)and Lemma 4.3,we obtain thatuμ+νμ∈C([0,T],H?α(?)).Then,

But from(4.4),we have:

Therefore,uμ(0)+νμ(0)=u0+ν0.By Theorem 4.1 in[12],we haveuμ(0)=u0,νμ(0)=ν0.

Theorem 4.4 generalizes the result of the global existence of weak solution for the standard nonlinear Schr¨odinger equation in[10].Now we prove our main result.

By a priori estimates in Lemmas 3.1-3.6 and Theorem 4.4,we derive a global smooth solutionu(t,x),ν(t,x)of the problem(1.1)-(1.3)[12],such that

Now we come to prove the uniqueness of the solution to the problem(1.1)-(1.3).

Suppose that(u1,ν1)and(u2,ν2)are two solutions which satisfy problem(1.1)-(1.3). Thenw1=u1?u2,w2=ν1?ν2satisfies

withw1(0)=w2(0)=0.

Taking the inner product of equations(4.16)withw1,w2,we obtain

Consider the imaginary part of(4.17)and then

In the same way,we can get

By the above inequalities,Gronwall’s inequality and(3.13),it follows from[12,p.709]that

Therefore,we complete the proof of Theorem 2.5.

Acknowledgement

This research is partly supported by the Hujiang Foundation of China(B14005)and NNSFC(11471215).

References

[1]Feynman R P,Hibbs A R.,Quantum Mechanics and Path Integrals,McGraw-Hill,New York, 1965.

[2]Laskin N.,Fractional quantum mechanics and L′evy integrals.Phys.Lett.A268(2000),298-305.

[3]Laskin N.,Fractional quantum mechanics.Phys.Rev.E63(2000),3135.

[4]Laskin N.,Fractional Schr¨odinger equation.Phys.Rev.E66(2002),056108.

[5]Guo Xiaoyi,Xu Mingyu,Some physical applications of fractional Schr¨odinger equation.J. Math.Phys.47(2006),082104.

[6]Guo Boling,Wang Lireng,Periodic Solutions for Nonlinear Systems of Schr¨odinger Equations and Kdv Equation.Journal of East China Normal University Natural Science Edition4(1986),42-48.

[7]Guo Boling,The existence and nonexistece of a global solution for one class of strong nonlinear schr¨odinger equations of fourth order.Applied Mathematics a journal of Chinese Universities6(2)(1991),173-182.

[8]Guo Boling,The initial problem with integral nonlinear schr¨odinger system of equations.Acta Mathematica Scientia1(3-4)(1981),261-274.

[9]Guo Boling,A class of high multidimensional nonlinear systems of schr¨odinger equations for initial boundary value and the periodic initial value problem.Chinese Scinence Bulletin6(1982),324-327.

[10]Guo Boling,Wang Lireng,Some Methods of Nonlinear Boundary Value Problem(Chinese translation),Zhongshan University Publishing House,1989.

[11]Guo Boling,Nonlinear Evolution Equations(Chinese),ShanghaiScientific and Technological Education Publishing House,Shanghai,1995.

[12]Guo Boling,Han Yongqian and Xin Jie,Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schr¨odinger equation.Applied Mathematics and Computation204(2008),468-477.

[13]Lawrence C.Evans,Partial Differential Equations,Second edition,American Mathematical Society providence,Rhode island,2010,275-280,208-209.

[14]Xuan Benjin,Variational Methods-Theory and Applications,University of Science and Technology of China Publishing House,2006,260-261.

[15]Liu Bingchu,Functional Analysis,second edition(Chinese),Science Publishing House,2004.

Received 4 September 2014;Accepted 24 March 2015

?Corresponding author.Email addresses:gmweixy@163.com(G.M.Wei),dongjing1666@126.com(J.Dong)

AMS Subject Classifications:35A01,35Q55

Chinese Library Classifications:O175.29