An averaging theorem for nonlinear Schr\"odinger equations with small nonlinearities

Consider nonlinear Schr\"odinger equations with small nonlinearities \[\frac{d}{dt}u+i(-\triangle u+V(x)u)=\epsilon \mathcal{P}(\triangle u,u,x),\quad x\in \mathbb{T}^d.\eqno{(*)}\] Let $\{\zeta_1(x),\zeta_2(x),\dots\}$ be the $L_2$-basis formed by eigenfunctions of the operator $-\triangle +V(x)$. For any complex function $u(x)$, write it as \mbox{$u(x)=\sum_{k\geqslant1}v_k\zeta_k(x)$} and set $I_k(u)=\frac{1}{2}|v_k|^2$. Then for any solution $u(t,x)$ of the linear equation $(*)_{\epsilon=0}$ we have $I(u(t,\cdot))=const$. In this work it is proved that if $(*)$ is well posed on time-intervals $t\lesssim \epsilon^{-1}$ and satisfies there some mild a-priori assumptions, then for any its solution $u^{\epsilon}(t,x)$, the limiting behavior of the curve $I(u^{\epsilon}(t,\cdot))$ on time intervals of order $\epsilon^{-1}$, as $\epsilon\to0$, can be uniquely characterized by solutions of a certain well-posed effective equation.


Introduction
We consider the Schrödinger equation where P : C d+2 × T d → C is a smooth function, 1 V (x) ∈ C n (T d ) is a potential (we will assume that n is sufficiently large) and ǫ ∈ (0, 1] is the perturbation parameter. For any p ∈ R denote by H p the Sobolev space of complex-valued periodic functions, provided with the norm || · || p , where ·, · is the real scalar product in L 2 (T d ), If p > d 2 + 2 = p d , then the mapping H p → H p−2 , u(x) → P(△u, ∇u, u, x) is smooth (see below Lemma 2.1). For any T > 0, a curve u ∈ C([0, T ], H p ), p > p d , is called a solution of (0.2) in H p if it is a mild solution of this equation. That is, if the relation obtained by integrating (0.2) in t from 0 to s holds for any 0 s T . We wish to study long-time behaviours of solutions for (0.2) and assume: Assumption A (a-priori estimate). Fix some T > 0. For any p > p d + 2, there exists n 1 (p) > 0 such that if n n 1 (p), then for any 0 < ǫ 1, the perturbed equation (0.2), provided with initial data has a unique solution u(t, x) ∈ H p such that ||u|| p C(T, p, ||u 0 || p ), for t ∈ [0, T ǫ −1 ].
Here and below the constant C also depends on the potential V (x). Denote the operator A V u := −△u + V (x)u. Let {ζ k } k 1 and {λ k } k 1 be its real eigenfunctions and eigenvalues, ordered in such a way that 1 λ 1 λ 2 · · · . We say that a potential V (x) is non-resonant if ∞ k=1 λ k s k = 0, (0. 4) for every finite non-zero integer vector (s 1 , s 2 , · · · ). For any complex-valued function u(x) ∈ H p , we denote by Abusing notation we will write v = (I, ϕ). Define h p I to be the weighted l 1 -space and consider the mapping It is continuous and its image is the positive octant h p I+ = {I ∈ h p I : I j 0, ∀j}. We mainly concern with the long time behavior of the actions I(u(t)) ∈ R ∞ + of solutions for the perturbed equation (0.2) for t ǫ −1 . For this purpose, it is convenient to pass to the slow time τ = ǫt and write equation (0.2) in the actionangle coordinates (I, ϕ): where I ∈ R ∞ , ϕ ∈ T ∞ and T ∞ := {(θ i ) i∈N : θ i ∈ T} is the infinite-dimensional torus endowed with the Tikhonov toppology. The functions F k and G k , k 1 represent the perturbation term P, 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 where I ∈ R M and ϕ ∈ T m , on time intervals of order ǫ −1 the action components I(t) can be well approximated by solutions of the following averaged equation: This assertion has been justified under various non-degeneracy assumptions on the frequency vector W and the initial data (I(0), ϕ(0)) (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): where dϕ is the Haar measure on T ∞ . But now, in difference with the finitedimensional case, the well-posedness of equation (0.10) is not obvious, since the map F (I) = ( F 1 (I), . . . ) is unbounded and the functions F k (I), k 1, may be not Lipschitz with respect to I in h p I+ . In [9], S. Kuksin observed that the averaged equation (0.10) may be lifted to a regular 'effective equation' on the variable v ∈ h p , which transforms to (0.10) under the projection π I . To derive an effective equation, corresponding to our problem, we first use mapping Ψ to write (0.2) as a system of equation on the vector v(τ ): (0.11) Here P (v) is the perturbation term P, written in v-variables. This equation is singular when ǫ → 0. The effective equation for (0.11) is a certain regular equatioṅ v = R(v). (0.12) To define the effective vector filed R(v), for any θ = (θ 1 , θ 2 , · · · ) ∈ T ∞ let us denote by Φ θ the linear operator in the space of complex sequences (v 1 , v 2 , · · · ) ∈ h p which multiplies each component v j with e iθj . Rotation Φ θ acts on vector fields on the v-space, and R(v) is the result of action of Φ θ on P (v), averaged in θ: The map R(v) is smooth with respect to v in h p . 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 p > p d + 2, there exists n 2 (p) > 0 such that if n n 2 (p), then for any initial data v 0 ∈ h p , there exists T (|v 0 | p ) > 0 such that the effective equations (0.12) has a unique solution v ∈ C([0, T (|v 0 | p )], h p ). Here T : R + → R >0 is an upper semi-continuous function.
The main result of this paper is the following statement, where v ǫ (τ ) is the Fourier transform of a solution u ǫ (t, x) for the problem (0.2), (0.3) (existing by Assumption A), written in the slow time τ = ǫt: We also assume Assumption B.
Proposition 0.2. The assumptions A and B hold if (0.2) is a complex Ginzburg-Landau equatioṅ where the constants γ R , γ I satisfy γ R , γ I > 0, (0.14) the functions f p (r) and f q (r) are the monomials |r| p and |r| q , smoothed out near zero, and 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: assuming the perturbation ǫf (u)(·) defines a smoothing mapping u(·) → f (u)(·). This additional assumption is necessary to guarantee the existence of an quasiinvariant 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 V (x) is non-degenerated and that the initial data u 0 (x) 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 O(ǫ −1 ) that we consider, precisely, over a time interval of order ǫ −m , with arbitrary m (even of order exp ǫ −δ with δ > 0 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 A V . 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.

As in the introduction,
and {λ k } k 1 are the eigenvalues of A V . According to Weyl's law, the λ k , k 1, satisfy the following asympototics Fix an L 2 -orthogonal basis of eigenfunctions {ζ k } k 1 corresponding to the eigenvalues {λ k } k 1 , and define the linear mapping Ψ as (0.5). For any m ∈ N, we have is C n -smooth, we have the following: The complement of E M is a real analytic variety in C n (T d ) of codimension at least 2, so E M is connected. The mapping Λ M is analytic in E M (see [8]).
Let µ be a Gaussian measure with a non-degenerate correlation operator, supported by the space C n (T d ) (see [4]). Then g. see [8]), then µ(Q s ) = 0 (see chapter 9 in [4] and the note [5]). Since this is true for any M and s as above, then we have: The non-resonant potentials form a subset of C n +1 (T d ) of full µ-measure.

Equation (0.2) in action-angle variables
For k = 1, 2, . . . , we denote: 2) (Here and below (·, ·) indicates the real scalar product in C, i.e. (u, v) = Re uv.) Denote Denoting for brevity, the vector field in equation (2.3) by ǫ −1 λ k + G k (v), we rewrite the equation for the pair (I k , ϕ k )(k 1) aṡ (Note that the second equation has a singularity when I k = 0.) We denote The following result is well known, see e.g. Section 5.5.3 in [14].
is C ∞ -smooth for p > d/2. Moreover, it is bounded and Lipschitz, uniformly on bounded subsets of H p (T d , C m ).
In the lemma below, P k and P j k are some fixed continuous functions. Lemma 2.2. For any j, k ∈ N, we have for any and for any m ∈ N and any 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.

Averaged equation and Effective equation
For a function f on a Hilbert space for a suitable continuous function P which depends on f . Clearly, the set of functions Lip loc (H) is an algebra. By Lemma 2.1, For a function f ∈ Lip loc (h p ) and any positive integer N , we define the average of f in the first N angles as and define the averaging in all angles as where dθ is the Haar measure on T ∞ . We will denote · ϕ as · when there is no confusion. The estimate (3.3) readily implies that Let v = (I, ϕ), then f N is a function independent of ϕ 1 , · · · , ϕ N , and f is independent of ϕ. Thus f can be written as f (I). Denote C 0+1 (T n ) the set of all Lipschitz functions on T n . The following result is a version of the classical Weyl theorem.
Lemma 3.2. Let f ∈ C 0+1 (T n ) for some n ∈ N. For any non-resonant vector ω ∈ R n (see (0.4)) and any δ > 0, there exists T 0 > 0 such that if T T 0 , g ∈ C(T n ) and |g − f | δ/3, then we have Proof. It is well known that for any δ > 0 and non-resonant vector ω ∈ R n , there exists T 0 > 0 such that (see e.g. Lemma 2.2 in [7]). Therefore if T T 0 , g ∈ C(T n ) and |g − f | δ/3, then This finishes the proof of the lemma.
Similar to equation (0.2), for any T > 0, we call a curve J ∈ C([0, T ], h p I ) a solution of equation (3.5) if for every s ∈ [0, T ] it satisfies the relation, obtained by integrating (3.5).
Consider the differential equationṡ v k = R k (v), k 1. Proof.
Applying Φ θ to (3.8) we get that Relation (3.7) implies that operations R and Φ θ commute. Therefore The assertion follows.

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 p p d + 2, assume n max{p, n 1 (p), n 2 (p)} and consider u 0 ∈ H p . So All constants below depend on M 1 (i.e. on M 0 ), and usually this dependence is not indicated. From the definition of the perturbation and Lemma 2.1 we know that where k = 1, · · · , n 0 . From now on, we always assume that (I, ϕ) ∈ B p (M 1 ) × T ∞ . Since V (x) is non-resonant, then by Lemma 2.2 and Lemma 3.2, for any ρ > 0, there exists T 0 = T 0 (ρ, n 0 ) > 0, such that for all ϕ ∈ T ∞ and T T 0 , where k = 1, . . . , n 0 . Due to Lemma 2.2, we have (4.7) From Lemma 3.1, we know and by (3.2), where Π I,ϕ (v m0 ) = (I m0 , ϕ m0 ) (see (2.5)) and | · | is the l ∞ -norm. Denote From now on we shall use the slow time τ = ǫt.
On each subsegment [a i , a i+1 ], we now consider the unperturbed linear dynamics ϕ i (τ ) of the angles ϕ m0 ∈ T m0 : Here the first inequality comes from equation (2.4), and using (4.7) we can get the second inequality. Therefore, using again (4.7), we have Therefore (4.15) holds for the same reason as (4.13).
We will now compare the integrals ai+1 ai This implies the inequality (4.16).
Finally, we have obvious Proposition 5.

Proof.
Due to (2.6) and (4.4), we know that for any ǫ ∈ (0, 1), Then by the Arzelà-Ascoli theorem, we have that the set I : ). Let {ρ m } m∈N be a sequence such that ρ m ց 0.
For any θ ∈ T ∞ and any vector I ∈ h p I+ we set where θ = (θ 1 , θ 2 , . . . ) and V θj (I j ) = 2I j cos(θ j ) + i 2I j sin(θ j ), for every j 1. Then ϕ j (V θj ) ≡ θ j , and for each θ ∈ T ∞ the map I → V θ (I) is a right inverse of the map v → I(v). For any vector I we denote  Proof.

application to complex Ginzburg-Landau equations
In this section we prove that assumptions A and B hold for equation (0.13), satisfying (0.14) and (0.15).

Verification of Assumption A.
In this subsection, we denote by | · | s the L s -norm. Let u(τ ) be a solution of equation (0.13) such that u(0, x) = u 0 . Then Since ||u|| 2 0 |u| 2 2p+2 , then relation ||u(τ 1 )|| 0 > γ So for any T > 0 we have Now we rewrite equation (0.13) as follows: . The l.h.s is a hamiltonian system with the hamiltonian function ǫ −1 H(u), We have dH(u)(v) = A V u, v + ǫγ I |u| 2q u, v , and if v is the vector field in the l.h.s of (5.2), then dH(u)(v) = 0. So we have and a similar relation holds for q replaced by p. Therefore where C 1 depends only on |V | C 2 . By this relation and (5.1), we have H(u(T )) H(u(0)) + C 1 T B 2 2 , for any T > 0.