AVERAGING

. Averaging principle for the cubic nonlinear Schr¨odinger equations with rapidly oscillating potential and rapidly oscillating force are obtained, both on ﬁnite but large time intervals and on the entire time axis. This in-cludes comparison estimate, stability estimate, and convergence result between nonlinear Schr¨odinger equation and its averaged equation. Furthermore, the existence of almost periodic solution for cubic nonlinear Schr¨odinger equations is also investigated.

1. Introduction.In this paper, we consider the averaging principle for the following Schrödinger equation where I = (0, 1), γ > 0, m is the rapidly oscillating potential function and f is the rapidly oscillating force function.We want to know the motion of large scale Schrödinger equation under rapidly oscillating potential and rapidly oscillating forcing.
The conservation form of this equation (γ = f = 0) has been extensively studied as a fundamental equation in modern mathematical physics (for a treatment in the whole space case see [26], and for periodic boundary case see [6] and the references therein).The equation in the form as presented in (1) has been derived from plasma physics (see [21] ) and as an amplitude equation in a perturbation study of the sine-Gordon equation (see [2]).
Our aim in this work is to obtain some information on the behavior of the solutions to the Schrödinger equation (1) as ε → 0. Starting from the fundamental work of Bogolyubov [3] the averaging theory for ordinary differential equations has been developed and generalized in a large number of works (see [4,12,20] and the references therein).Bogolyubov's main theorems have been generalized in [10] to 2148 PENG GAO AND YONG LI the case of differential equations with bounded operator-valued coefficients.Some problems of averaging of differential equations with unbounded operator-valued coefficients have been considered in [10,16,19,25] in the framework of abstract parabolic equations.[9] obtains a general version of the Bogolyubov averaging lemma for a time-varying differential equation.The works [11,17,18] are devoted to generalization of method of averaging for dissipative partial differential equations.[8] uses the concept of pullback attractors of such systems to establish the existence of almost periodic (quasi-periodic, almost automorphic, recurrent, pseudo recurrent) solutions corresponding to time dependent coefficients of these types and a global averaging principle is derived.[14] establishes averaging principle for quasigeostrophic motion.[13] derives an averaging principle for the 2D quasi-geostrophic flow.
Motivated by previous research and from both physical and mathematical standpoints, the following mathematical questions arise naturally which are important from the point of view of dynamical systems: Does the averaging principle for cubic nonlinear Schrödinger equations (1) with rapidly oscillating potential and rapidly oscillating force hold ?
In this paper we will answer the above question.We first rewrite (1) as if we set Au = i(u xx + iγu), N (u) = i|u| 2 u, we have its averaged equation is We set τ = t ε , (2) and (3) become Throughout this paper, we use the following notations: Let X be a Banach space and let C b ([0, +∞); X)(C b (R; X)) be the Banach space of all X-valued strongly continuous bounded functions defined on [0, +∞)(R), L p loc (R, X) be the Banach space of all X-valued function f satisfying We first prove the averaging principle on a finite time interval, the so-called first Bogolyubov theorem: Theorem 1.1.(Averaging on a finite time interval) We assume the potential function m and the force function f satisfy the following hypothesis: where M > 0 is a positive constant and σ(τ ) → 0 as τ → ∞. m 0 is a real function.
Let T > 0 be arbitrary and fixed, u and ū be the solutions of ( 4) and ( 5), 2. This theorem gives comparison estimate, stability estimate and convergence result (as ε → 0) between ( 4) and ( 5), on finite but large time intervals.
The second result in this paper is the averaging principle on the entire real axis, the so-called second Bogolyubov theorem: Theorem 1.2.(Averaging on the entire axis) We assume the potential function m and the force function f satisfy the following hypothesis: There exists sufficiently small δ 0 .

H2) (averaged condition)
There exist functions f 0 , m 0 ∈ H 3 (I) such that where M > 0 is a positive constant and σ(τ ) → 0 as τ → ∞.H3) There exists a function If ε is small enough, equation (4) has the following properties: 1) In a small neighbourhood of the stationary point u 0 , equation (4) has a unique solution u * (τ ), which is bounded on the entire axis and satisfies: 2) If the functions f, m : R → H 1 (I) are almost periodic functions, then the solution u * is almost periodic with frequency basis contained in that of m and f.
Remark 2. 1. H3) can be obtained from the classical regularity theory of elliptic equation.
2. This theorem shows that if there exists a stationary solution u 0 of the averaged equation and this stationary point is small enough, then in a small neighbourhood of this point there exists a solution u * (τ ) of the original equation that is bounded on the entire real axis and is conditionally exponentially stable.Moreover, this solution is almost periodic if m and f are almost periodic.
The main difficulties in the proofs of Theorem 1.1 and Theorem 1.2 are the wellposedness and regularity of Schrödinger equations in H 3 (I), we obtain these results in Proposition 4. In order to deal with the estimates in averaging principle, we should use H 3 (I).
This paper is organized as follows.In Section 2, we present some auxiliary results that are required for this paper.In Section 3, we prove Theorem 1.1.Section 4 provides the proof of Theorem 1.2.

2.
Preliminaries.In this section, we give some auxiliary results, which will be used in the proof of Theorem 1.1 and Theorem 1.2.
then u satisfies the following identical equations Let S(t) = e At be the semigroup corresponding to the equation Then, we have Proposition 2. If γ > 0, we have for any k ∈ N.
Proof.The case k = 0 can be obtained from Proposition 1. Set Au = iu xx − γu and then it is easy to see that the solution u satisfies By the same idea as in [5, P1425] and [23, P940, Proof of Proposition 2.7], set v = u kt , we have Namely, we have The odd case can be obtained from the interpolation argument.
Definition 2.1.(see [1]) A semigroup of continuous linear operators S(t) 2) the function S(t)ϕ is weakly continuous in t; 3) there exists a number r 0 , 0 < r 0 < 1, such that outside the circle |ζ| ≤ r 0 there are only finitely many points of the spectrum of the operator S(1); 4) the invariant subspace corresponding to this part of the spectrum is finitedimensional.
According to Proposition 2, we have Proof.The main idea comes from [17,P663,Lemma 5].According to Proposition 2 and the spectral radius theorem we see that the spectrum of S(t) is contained inside the disc |ζ| ≤ e −γt < 1.Hence, the essential spectrum σ e (S(t)) of the operator S(t) is contained inside the same disc disc |ζ| ≤ e −γt < 1.The operator S(t) is bounded; hence the connected component of the set C\σ e (S(t)) containing infinity contains the exterior of our disc, that is, the set |ζ| > e −γt .Therefore each point in the exterior of this disc either belongs to the resolvent set of the operator S(t) or is an isolated eigenvalue of it with finite multiplicity.Thus, outside this circle there are only finitely many eigenvalues counting multiplicities.Thus S(t) is almost stable in H 1 (I).
Let σ + (S(1)) = {ζ 1 , . . ., ζ n } and γ be a contour in the resolvent set enclosing σ + (S(1)).We denote by P + the spectral projection and denote by P − the complementary projection The subspaces H + = P + H 1 (I) and H − = P − H 1 (I) are invariant with respect to S(t) for all t: We set for t = 0 Then, we have the following result: Proposition 3. The inhomogeneous equation ) bounded on the entire axis: Moreover, there exist δ ∈ (0, 1) and K such that where Proof.It follows from [1, Theorem IV.2.1] that there exist δ ∈ (0, 1) and K such that S(t)P − L(H 1 (I);H 1 (I)) ≤ Kδ t t > 0, In order to ensure the rationality of the hypothesis H3) in Theorem 1.1, we prove the following proposition: admits a solution u ∈ C([0, +∞), H 3 (I) H 1 0 (I)), moreover, for any t > 0, u satisfies the following estimate where C is independent of t.
Proof.According to Proposition 2, the classical argument and the Banach contraction principle, we can obtain the local well-posedness of (8).
According to Proposition 1, we have 1 2 Step 2. Estimate of u H 1 (I) .
By the same method as in [15] and [27], we can obtain that 1 2 we have 1 2

PENG GAO AND YONG LI
We set Noting that Combining ( 10) and ( 11), we deduce that a direct computation shows that Together ( 9), ( 11) and (12), we arrive at where C is independent of t.Namely, Step 3. We should prove according to Gagliardo-Nirenberg inequality, we have   L ∞ (I) , thus, According to [24] or [7, P678,Lemma 4], we shall show T max = +∞ by proving that u H 3 (I) remains bounded on every finite time interval.
Due to the semigroup theory, we have By the same method as in [7, Lemma 2], we can obtain the following result: For any u ∈ H 2 (I) with u H 1 (I) ≤ 1, we have where C is a constant depending only on I.
The proof of this assertion is obtained by slight modification of arguments from [7, Lemma 2] and we omit the details We define Au = u xxx , then Combining ( 9) and ( 13), we deduce that Setting G(t) = the right-hand sides of ( 14), we have Therefore u(t) H 3 (I) remains bounded on every finite time interval and so we must have T max = +∞, thus there exists a unique solution of (8) such that u ∈ C([0, +∞), H 3 (I) H 1 0 (I)).
3. Proof of Theorem 1.1.It was shown in [27, P234], that for m 0 ∈ H 1 (I) the semigroup F (t) corresponding to equation (3) possesses absorbing sets in the space H 1 (I), in particular, the ball B H 1 (I) (R 0 ), where R 0 is large enough.This means that for every bounded set B, In addition, the semigroup is uniformly bounded in these spaces, that is, given any ball, in particular, the ball B H 1 (I) (R 0 ), there exists a ball B H 1 (I) (R) such that By increasing R we may assume that We suppose that the trajectory of the initial equation with u(0 ].This will be proved in the end of this section.Given a point u 0 in B H 1 (I) (R 0 ), let trajectories of systems ( 4) and ( 5) start from this point.Set z = u − ū, then z satisfies By the semigroup theory, we can obtain the following equivalent integral equation we have, Let us estimate the first term in the right-hand side of ( 15) Next, we estimate the second term in the right-hand side of (15), integrating by parts, we have At last, we estimate the term τ 0 e εA(τ −s) [i(f − f 0 )]ds, integrating by parts, by the same method as in the above estimates, we have By the same method as in [17, P659], according to the above inequalities, we have where G(ε) → 0 as ε → 0. It follows from Gronwall inequality that Namely, by assuming that the trajectory u(t) with initial value u(0) ∈ B H 1 (I) (R 0 ) stays in the ball B H 1 (I) (R) on the interval [0, T ε ], we have proved the proximity of solutions of ( 4) and ( 5) in H 1 (I).
Next, we prove that the trajectory u(t) with initial condition u(0) ∈ B H 1 (I) (R 0 ) stays in the ball B H 1 (I) (R) on the interval [0, T  ε ].Indeed, let ε be so small that the right-hand side of ( 16) is less than ρ 2 , where ρ is defined earlier in this section when we discuss absorbing sets.Suppose that the trajectory u(t) leaves the ball B(R) during the interval [0, T ε ] and let τ * be the first moment when u(τ * ) H 1 (I) = R.However, on the interval τ ∈ [0, τ * ] both trajectories stay in the ball B H 1 (I) (R) and what we have proved so far shows that the inequality u(τ ) − ū(τ ) H 1 (I) ≤ ρ 2 is valid.In particular, it is valid for τ = τ * .This together with the inequality ū(τ * ) H 1 (I) ≤ R − ρ, which holds by the hypothesis of the theorem, gives the contradiction: 4. Proof of Theorem 1.2.Note the fact We change the variable in the equation u = u 0 + h − εv in ( 4), where v and h satisfy and we set 4.1.Study of v.We consider equation (17).
Proposition 5. Equation ( 17) has a unique solution v(τ, ε) bounded in H 1 (I) uniformly in τ ∈ R. Moreover If m and f are almost periodic with values in H 1 (I), then v is almost periodic in H 1 (I) with frequency basis contained in that of m and f .
Proof.The desired solution is given by the formula It follows from Proposition 2 that It is easy to know that for any δ > 0, there exists a s 0 = s 0 (δ) so large that σ(s) < δ when s > s 0 .Thus, integrating by parts, it holds that Integrating by parts yields It follows from ( 19) and ( 21) that Let ε be small enough such that we have sup Letting δ → 0 and then ε → 0 we obtain lim Let us prove the last statement of the proposition.
It is easy to check that and sup It follows from the above inequalities that sup Let ε be small enough, we have sup it is sufficient to show that every m-recurrent sequence {τ m } is also v-recurrent.

4.2.
Study of h.We revert to the original time t in ( 18) where Proposition 6.Let ε, δ 0 be small enough, equation ( 22) has a unique bounded solution h * with the following properties: is almost periodic, then solution h * is almost periodic with frequency basis contained in that of Q.
Proof. 1) Some properties of function Q are given below.A simple calculation yields If h 1 , h 2 ∈ B H 1 (I) (ρ) and choosing ρ, ε, u 0 H 1 (I) , m C b (R,H 1 (I)) small enough, we can obtain that F is a contraction map taking a ball in C b (R, H 1 (I)) of radius ρ into itself.By the Banach contraction principle, F has a unique fixed point Fh * = h * , in other words, h * is a H 1 (I)-bounded solution of equation (22).