Averaging theorem for nonlinear Strödinger equations

# An averaging theorem for nonlinear Schrödinger equations with small nonlinearities

## Abstract.

Consider nonlinear Schrödinger equations with small nonlinearities

 ddtu+i(−△u+V(x)u)=ϵP(△u,u,x),x∈Td. *

Let be the -basis formed by eigenfunctions of the operator . For any complex function , write it as and set . Then for any solution of the linear equation we have . In this work it is proved that if is well posed on time-intervals and satisfies there some mild a-priori assumptions, then for any its solution , the limiting behavior of the curve on time intervals of order , as , can be uniquely characterized by solutions of a certain well-posed effective equation.

## Introduction

We consider the Schrödinger equation

 ddtu+i(−△u+V(x)u)=0,x∈Td, (0.1)

and its nonlinear perturbation:

 ddtu+i(−△u+V(x)u)=ϵP(△u,∇u,u,x),x∈Td, (0.2)

where is a smooth function, is a potential (we will assume that is sufficiently large) and is the perturbation parameter. For any denote by the Sobolev space of complex-valued periodic functions, provided with the norm ,

 ||u||2p=⟨(−△)pu,u⟩+⟨u,u⟩,ifp∈N,

where is the real scalar product in ,

 ⟨u,v⟩=Re∫Tdu¯vdx,u,v∈L2(Td).

If , then the mapping , is smooth (see below Lemma 2.1). For any , a curve , , is called a solution of (0.2) in if it is a mild solution of this equation. That is, if the relation obtained by integrating (0.2) in from to holds for any . We wish to study long-time behaviours of solutions for (0.2) and assume:

Assumption A (a-priori estimate). Fix some . For any , there exists such that if , then for any , the perturbed equation (0.2), provided with initial data

 u(0)=u0∈Hp, (0.3)

has a unique solution such that

 ||u||p⩽C(T,p,||u0||p),fort∈[0,Tϵ−1].

Here and below the constant also depends on the potential .

Denote the operator

 AVu:=−△u+V(x)u.

Let and be its real eigenfunctions and eigenvalues, ordered in such a way that

 1⩽λ1⩽λ2⩽⋯.

We say that a potential is non-resonant if

 ∞∑k=1λksk≠0, (0.4)

for every finite non-zero integer vector . For any complex-valued function , we denote by

 Ψ(u):=v=(v1,v2,⋯),vj∈C, (0.5)

the vector of its Fourier coefficients with respect to the basis , i.e. . In the space of complex sequences , we introduce the norms

 |v|2p=∑k⩾1|vk|2λpk,p∈R,

and define . Denote

 Ik=12|vk|2,φk=Argvk,k⩾1. (0.6)

Then are the action-angles for the linear equation (0.1). That is, in these variables equation (0.1) takes the integrable form

 ddtIk=0,ddtφk=λk,k⩾1. (0.7)

Abusing notation we will write . Define to be the weighted -space

 hpI:={I=(I1,…)∈R∞:|I|∼p<+∞},|I|∼p=2∞∑i=1λpi|Ii|,

and consider the mapping

 πI:hp→hpI,v↦I,Ij(v)=12|vj|2,j⩾1.

It is continuous and its image is the positive octant .

We mainly concern with the long time behavior of the actions of solutions for the perturbed equation (0.2) for . For this purpose, it is convenient to pass to the slow time and write equation (0.2) in the action-angle coordinates :

 ˙Ik=Fk(I,φ),˙φk=ϵ−1λk+Gk(I,φ),k⩾1, (0.8)

where , and is the infinite-dimensional torus endowed with the Tikhonov toppology. The functions and , represent the perturbation term , written in the action-angle coordinates. In the finite dimensional situation, the averaging principle is well established for perturbed integrable systems. The principle states that for equations

 ddtI=ϵf(I,φ),ddtφ=W(I)+ϵg(I,φ),

where and , on time intervals of order the action components can be well approximated by solutions of the following averaged equation:

 ddtJ=ϵ⟨f⟩(J),⟨f⟩(J)=∫Tmf(J,φ)dφ. (0.9)

This assertion has been justified under various non-degeneracy assumptions on the frequency vector and the initial data (see [12]). In this paper we want to prove a version of the averaging principle for the perturbed Schrödinger equation (0.2). We define a corresponding averaged equation for (0.8) as in (0.9):

 Extra open brace or missing close brace (0.10)

where is the Haar measure on . But now, in difference with the finite-dimensional case, the well-posedness of equation (0.10) is not obvious, since the map is unbounded and the functions , , may be not Lipschitz with respect to in . In [9], S. Kuksin observed that the averaged equation (0.10) may be lifted to a regular ‘effective equation’ on the variable , which transforms to (0.10) under the projection . To derive an effective equation, corresponding to our problem, we first use mapping to write (0.2) as a system of equation on the vector :

 ˙v=ϵ−1dΨ(u)(−iAV(u))+P(v). (0.11)

Here is the perturbation term , written in -variables. This equation is singular when . The effective equation for (0.11) is a certain regular equation

 ˙v=R(v). (0.12)

To define the effective vector filed , for any let us denote by the linear operator in the space of complex sequences which multiplies each component with . Rotation acts on vector fields on the -space, and is the result of action of on , averaged in :

 R(v)=∫T∞Φ−θP(Φθv)dθ.

The map is smooth with respect to in . Again, we understand solutions for equation (0.12) in the mild sense.

We now make the second assumption:

Assumption B (local well-posedness of the effective equation). For any , there exists such that if , then for any initial data , there exists such that the effective equations (0.12) has a unique solution . Here is an upper semi-continuous function.

The main result of this paper is the following statement, where is the Fourier transform of a solution for the problem (0.2), (0.3) (existing by Assumption A), written in the slow time :

 vϵ(τ)=Ψ(uϵ(ϵ−1τ)),τ∈[0,T].

We also assume Assumption B.

###### Theorem 0.1.

For any , if , then there exists such that for every ,

 I(vϵ(⋅))⟶ϵ→0I0(⋅)inC([0,T],hqI).

Moreover , , solves the averaged equation (0.10) with initial data , and it may be written as , where is the unique solution of the effective equation (0.12), equal to at .

###### Proposition 0.2.

The assumptions A and B hold if (0.2) is a complex Ginzburg-Landau equation

 ˙u+ϵ−1i(△u+V(x)u)=△u−γRfp(|u|2)u−iγIfq(|u|2)u,x∈Td, (0.13)

where the constants , satisfy

 γR,γI>0, (0.14)

the functions and are the monomials and , smoothed out near zero, and

 0⩽p,q<∞ifd=1,2and0⩽p,q

This work is a continuation of the research started in [7], where the author proved a similar averaging principle (not for all but for typical initial data) for a perturbed KdV equation:

 ut+uxxx−6uux=ϵf(u)(x),x∈T,∫Tu(t,x)dx=0, (0.16)

assuming the perturbation defines a smoothing mapping . This additional assumption is necessary to guarantee the existence of an quasi-invariant measure for the perturbed equation (0.16), which plays an essential role in the proof due to the non-linear nature of the unperturbed equation. Since in the present paper we deal with perturbations of a linear equation, this restriction is not needed.

In [10], a result similar to Theorem 0.1 was proved for weakly nonlinear stochastic CGL equation (0.13). There are many works on long-time behaviors of solutions for nonlinear Schrödinger equations. E.g. the averaging principle was justified in [8] for solutions of Hamiltonian perturbations of (0.1), provided that the potential is non-degenerated and that the initial data is a sum of finitely many Fourier modes. Several long-time stability theorems which are applicable to small amplitude solutions of nonlinear Schrödinger equations were presented in [1, 3, 13, 6]. The results in these works describe the dynamics over a time scale much longer than the that we consider, precisely, over a time interval of order , with arbitrary (even of order with in [1, 13, 6]). These results are obtained under the assumption that the frequencies are completely resonant or highly non-resonant (Diophantine-type), by using the normal form techniques near an equilibrium (this is the reason for which they only apply to small amplitude solutions). See [2] and references therein for general theory of normal form for PDEs. In difference with the mentioned works, the research in this paper is based on the classical averaging method for finite dimensional systems, characterizing by the existence of slow-fast variables. It deals with arbitrary solution of equation (0.2) with sufficiently smooth initial data. Also note that the non-resonance assumption (0.4) is significantly weaker than those in the mentioned works.

Plan of the paper. In Section 1 we recall some spectral properties of the operator . Section 2 is about the action-angle form of the perturbed linear Schrödinger equation (0.2). In Section 3 we introduce the averaged equation and the corresponding effective equation. Theorem 0.1 and Proposition 0.2 are proved in Section 4 and Section 5.

## 1. Spectral properties of Av

As in the introduction, , , where and are the eigenvalues of . According to Weyl’s law, the , , satisfy the following asympototics

 λk=Cdk2/d+o(k2/d),k⩾1,

Fix an -orthogonal basis of eigenfunctions corresponding to the eigenvalues , and define the linear mapping as (0.5). For any , we have , where . Noting that is equivalent to for , since is -smooth, we have the following:

###### Lemma 1.1.

For every integer the linear mapping is an isomorphism.

We denote

 Cn+1(Td):={V(x)⩾1:V(x)∈Cn(Td)}.

For any finite consider the mapping

 ΛM:Cn+1(Td)→RM,V(x)→(λ1,⋯,λM),

and define the open domain ,

 EM:={V|λ1<λ2<⋯<λM}.

The complement of is a real analytic variety in of codimension at least 2, so is connected. The mapping is analytic in (see [8]).

Let be a Gaussian measure with a non-degenerate correlation operator, supported by the space (see [4]). Then . Fix . The set

 Qs:={V∈EM|ΛM(V)⋅s=0},

is closed in . Since on (e.g. see [8]), then (see chapter 9 in [4] and the note [5]). Since this is true for any and as above, then we have:

###### Proposition 1.2.

The non-resonant potentials form a subset of of full -measure.

## 2. Equation (0.2) in action-angle variables

For , we denote:

 Ψk:Hp→C,Ψk(u)=vk,

(see (0.5)). Let be a solution of equation (0.2). Passing to slow time , we get for equations

 ˙vk+iϵ−1λkvk=Ψk(P(△u,∇u,u,x)),k⩾1. (2.1)

Since is an integral of motion for the Schrödinger equation (0.1), we have

 ˙Ik=(Ψk(P(△u,∇u,u,x)),vk):=Fk(v),k⩾1 (2.2)

(Here and below indicates the real scalar product in , i.e. .)

Denote , if , and , if , . Using equation (2.1), we get

 ˙φk=ϵ−1λk+|vk|−2(Ψk(P(△u,∇u,u,x)),ivk),ifvk≠0,k⩾1 (2.3)

Denoting for brevity, the vector field in equation (2.3) by , we rewrite the equation for the pair as

 ˙Ik=Fk(v)=Fk(I,φ),˙φk=ϵ−1λk+Gk(v). (2.4)

(Note that the second equation has a singularity when .) We denote

 F(I,φ)=(F1(I,φ),F2(I,φ),⋯).

The following result is well known, see e.g. Section 5.5.3 in [14].

###### Lemma 2.1.

If is , then the mapping

 Mf:Hp(Td,Cm)→Hp(Td,CN),u↦f(u),

is -smooth for . Moreover, it is bounded and Lipschitz, uniformly on bounded subsets of .

In the lemma below, and are some fixed continuous functions.

###### Lemma 2.2.

For any , we have for any

(i)The function is smooth in each space .

(ii) For any , the function is bounded by .

(iii)For any , the function is bounded by .

(iv) The function is bounded by and for any and any , the fucntion is smooth on .

###### Proof.

Item (i) and (ii) follow directly from (2.2), (2.3), Lemmata 1.1 and 2.1. Item (iii) and (iv) follow directly from item (i) and the chain rule. ∎

Denote

 ΠI,φ:hp→hpI×T∞,ΠI,φ(v)=(I(v),φ(v)). (2.5)
###### Definition 2.3.

Let assumption A holds. Then for any and , we call a curve , , a regular solution of equation (2.4), if there is a solution of equation (0.2) such that

 ΠI,φ(Ψ(u(ϵ−1τ)))=(I(τ),φ(τ))∈hpI×T∞,τ∈[0,T].

Note that if is a regular solution, then each is a -function, while may be discontinuous at points , where .

For any , let be a regular solution of (2.4) such that . Then by assumption A, for any and , we have

 |I(τ)|∼p=12|v(p)|2p⩽C(p,M0,T),t∈[0,T]. (2.6)

## 3. Averaged equation and Effective equation

For a function on a Hilbert space , we write if

 |f(u1)−f(u2)|⩽P(R)||u1−u2||,if||u1||,||u2||⩽R, (3.1)

for a suitable continuous function which depends on . Clearly, the set of functions is an algebra. By Lemma 2.1,

 Fk(v)∈Liploc(hp),k∈N,p>pd. (3.2)

Let and , where . Denoting by the projection

 ΠM:h0↦h0,(v1,v2,⋯)↦(v1,⋯,vM,0,⋯),

we have

 |v−ΠMv|p⩽λ−(p1−p)/2M|v|p1.

Accordingly,

 |f(v)−f(ΠMv)|⩽P(|v|p)λ−(p1−p)/2M|v|p1. (3.3)

We will denote and identify with if needed. Similar notations will be used for vectors and vectors .

The torus acts on the space by linear transformations , , where . Similarly, the tous acts on by linear transformations with .

For a function and any positive integer , we define the average of in the first angles as

 ⟨f⟩N(v)=∫TNf((ΦθN⊕id)(v))dθN,

and define the averaging in all angles as

 ⟨f⟩φ(v)=∫T∞f(Φθ(v))dθ,

where is the Haar measure on . We will denote as when there is no confusion. The estimate (3.3) readily implies that

 |⟨f⟩N(v)−⟨f⟩(v)|⩽P(R)λ−(p1−p)/2N,if|v|p1⩽R.

Let , then is a function independent of , and is independent of . Thus can be written as .

###### Lemma 3.1.

(See [11]). Let , then

1. Functions and satisfy (3.1) with the same function as and take the same value at the origin.

2. They are smooth if is. If is -smooth, then for any , is a smooth function of the first components of the vector .

###### Proof.

Item (i) and the first statement of item (ii) is obvious. Notice that is even on each variable , , i.e.

 ⟨f⟩(…,−√Ij,…)=⟨f⟩(…,√Ij,…),j⩾1.

Now the second statement of item (ii) follows from Whitney’s theorem (see Lemma A in the Appendix). ∎

Denote the set of all Lipschitz functions on . The following result is a version of the classical Weyl theorem.

###### Lemma 3.2.

Let for some . For any non-resonant vector (see (0.4)) and any , there exists such that if , and , then we have

 ∣∣1T∫T0g(x0+ωt)dt−⟨g⟩∣∣⩽δ,

uniformly in .

###### Proof.

It is well known that for any and non-resonant vector , there exists such that

 ∣∣1T∫T0f(x0+ωt)dt−⟨f⟩∣∣⩽δ/3,∀T⩾T0,

(see e.g. Lemma 2.2 in [7]). Therefore if , and , then

 ∣∣1T∫T0g(x0+ωt)dt−⟨g⟩∣∣⩽∣∣1T∫T0f(x0+ωt)dt−⟨f⟩∣∣+1T∫T0|f(x0+ωt)−g(x0+ωt)|dt+|⟨f⟩−⟨g⟩|⩽δ.

This finishes the proof of the lemma. ∎

We denote , then equations (2.4) becomes

 ˙Ik=(vk,Pk(v)),˙φk=ϵ−1λk+Gk(v),k⩾1. (3.4)

The averaged equations have the form

 ˙Jk=⟨(vk,Pk)⟩φ(J),k⩾1, (3.5)

i.e.

 ⟨(vk,Pk)⟩φ=∫T∞(vkeiθk,Pk(Φθv))dθ=(vk,Rk(v)), (3.6)

with

 Rk(v)=∫TΦ−θkPk(Φθ)dθ. (3.7)

Similar to equation (0.2), for any , we call a curve a solution of equation (3.5) if for every it satisfies the relation, obtained by integrating (3.5).

Consider the differential equations

 ˙vk=Rk(v),k⩾1. (3.8)

Solutions of this system are defined similar to that of (0.2) and (3.5). Relation (3.6) implies:

###### Lemma 3.3.

If satisfies (3.8), then satisfies (3.5).

Following [9], we call equations (3.8) the effective equation for the perturbed equation (0.2).

###### Proposition 3.4.

The effective equation is invariant under the rotation . That is, if is a solution of (3.8), then for each , also is a solution.

###### Proof.

Applying to (3.8) we get that

 ddτΦθv=ΦθR(v).

Relation (3.7) implies that operations and commute. Therefore

 ddτΦθv=R(Φθv).

The assertion follows. ∎

## 4. Proof of the Averaging theorem

In this section we prove the Theorem 0.1 by studying the behavior of regular solutions of equation (2.4). We fix , assume and consider . So

 ΠI,φ(Ψ(u0))=(I0,φ0)∈hpI+×T∞. (4.1)

We denote

 Bp(M)={I∈hpI+:|I|∼p⩽M}. (4.2)

Without loss of generality, we assume . Fix any . Let

 (I0,φ0)∈Bp(M0)×T∞:=Γ0,

and let be a regular solution of system (2.4) with . Then by (2.6), there exists such that

 I(τ)∈Bp(M1),τ∈[0,1]. (4.3)

All constants below depend on (i.e. on ), and usually this dependence is not indicated. From the definition of the perturbation and Lemma 2.1 we know that

 |F(I,φ)|∼p−2⩽CM1,∀(I,φ)∈Bp(M1)×T∞. (4.4)

Recall that we identify with , etc.

Fix any . By (3.2), for every , there is , depending only on , and , such that if , then

 |Fk(I,φ)−Fk(Im,φm)|⩽ρ,∀(I,φ)∈Bp(M1)×T∞, (4.5)

where .

From now on, we always assume that .

Since is non-resonant, then by Lemma 2.2 and Lemma 3.2, for any , there exists , such that for all and ,

 ∣∣1T∫T0Fk(Im0,φm0+Λm0t)dt−⟨Fk⟩(Im0)∣∣<ρ, (4.6)

where . Due to Lemma 2.2, we have

 |Gj(I,φ)|⩽C0(j,M1)√Ij,ifIj≠0,|∂Fk∂Ij(I,φ)|⩽C0(k,j,M1)√Ij,ifIj≠0,|∂Fk∂φj(I,φ)|⩽C0(k,j,M1). (4.7)

From Lemma 3.1, we know

 |⟨Fk⟩(Im0)−⟨Fk⟩(¯Im0)|⩽C1(k,m0,M1)|Im0−¯Im0|, (4.8)

and by (3.2),

 |Fk(Im0,φm0)−Fk(¯Im0,¯φm0)|⩽C2(k,m0,M1)|vm0−¯vm0|, (4.9)

where (see (2.5)) and is the -norm. Denote