ON THE EXISTENCE OF FULL DIMENSIONAL KAM TORUS FOR FRACTIONAL NONLINEAR SCHRÖDINGER EQUATION

In this paper, we study fractional nonlinear Schrödinger equation (FNLS) with periodic boundary condition iut = −(−∆)0u− V ∗ u− εf(x)|u|u, x ∈ T, t ∈ R, s0 ∈ ( 1 2 , 1), (0.1) where (−∆)0 is the Riesz fractional differentiation defined in [21] and V ∗ is the Fourier multiplier defined by V̂ ∗ u(n) = Vnû(n), Vn ∈ [−1, 1] , and f(x) is Gevrey smooth. We prove that for 0 ≤ |ε| ≪ 1 and appropriate V , the equation (0.1) admits a full dimensional KAM torus in the Gevrey space satisfying 1 2 e−rn θ ≤ |qn| ≤ 2e−rn θ , θ ∈ (0, 1), which generalizes the results given by [8–10] to fractional nonlinear Schrödinger equation.


Introduction and main results
In this paper, we focus on the fractional nonlinear Schrödinger equation (FNLS) with periodic boundary conditions where V * is a Fourier multiplier defined by and f (x) is Gevrey smooth. Written in Fourier modes (q n ) n∈Z , then (1.1) can be rewritten asq with the Hamiltonian H(q, q) = n∈Z (C(s 0 )n 2s0 + V n )|q n | 2 + ϵ n∈Z n1−n2+n3−n4 +n5−n6=−n f (n)q n1 q n2 q n3 q n4 q n5 q n6 , (1.3) where C(s 0 ) = − 1 cos s0π and n 2s0 = (n 2 ) s0 . The details about eigenvalues and eigenfunctions of operator (−∆) s0 have been carefully calculated by Li in [26]. Our aim is to show the existence of almost periodic solutions for such a family of FNLS.
For the best understanding of FNLS, we firstly consider the following equation (s 0 = 1 4 ) with periodic boundary condition iu t − |∂ x | 1 2 u + V * u + ϵf (|u| 2 )u = 0, x ∈ T, (1.4) where V * is a Fourier multiplier and f is real analytic near 0 ∈ C. The model is motivated by the water wave problem posed in a fluid of infinite depth by Craig and Worfolk [12] and Zakharov [33]. We also mention that in the physics literature the fractional Schrödinger equation was introduced by Laskin [28] to describe frational quantum mechanics. From then on, there have been many works on FNLS, see [14,19,28] for more details. More recently, by using KAM technique, Li in [26] showed that there were many quasi-periodic solutions of a class of space fractional nonlinear Schrödinger equations (1/2 < s 0 < 1) with the Riesz fractional differentiation and Xu in [32] obtained a family of small-amplitude quasi-periodic solutions with linear stability for FNLS (1.4). In fact, the KAM technique is a powerful tool to obtain quasi-periodic (or almost-periodic) solutions for Hamiltonian partial differential equations (PDEs). The KAM results, such as the existence results of quasi-periodic solutions for Hamiltonian PDEs have attracted a great deal of attention over years and is well understood in [1, 2, 5-7, 11, 13, 15, 20, 22-25, 27, 34] and references therein. In the all above works, the obtained KAM tori are lower (finite) dimension.
In this paper, we are interested in the construction of almost periodic solutions of (1.1), we will investigate the full dimensional tori by proceeding along the 'usual' KAM scheme where the perturbation is eventually removed by consecutive symplectic transformations of phase space. There are some papers focusing on the existence of the full dimensional KAM tori for Hamiltonian PDEs. The first related result is given by Fröhlich-Spencer-Wayne in [16] and Pöschel [30], where the infinite dimensional Hamiltonian system with short range was considered. Such infinitedimensional Hamiltonian systems are well approximated by finite-dimensional ones and one can show that the classical KAM proof also works in this case, if we choose proper norm. Later, the almost periodic solutions on a full set frequencies for one dimensional NLS and NLW were constructed by Bourgain in [4] (Also see the almost periodic solutions for one dimensional NLS in [17,18,31] and higher-dimensional beam equations by Niu-Geng in [29]). These invariant tori were obtained by imposing hyper-exponentially decay on actions I n (I n ∼ e −|n| S , S > 1 ). An open problem raised by Kuksin is whether there exist the full dimensional tori with suitable decay, such as I n ∼ |n| −S . The first result in this direction for Hamiltonian PDEs was given by Bourgain who in [8] proved that 1-dimensional NLS has a full dimensional KAM torus of prescribed frequencies where the actions of the torus obey the estimates with θ = 1/2. Recently, Bourgain's results have been extended to any θ ∈ (0, 1) in [9] and to the case that the nonlinear perturbation depends explicitly on the space variable x in [10]. Biasco-Massetti-Procesi in [3] generalized and rederive Bourgain's results in [8] with a more geometric point of view and construct many elliptic tori independent of their dimension. Indeed, our work was initiated and inspired by Bourgain and it turns out the KAM scheme is still applicable in (1.1) due to special arithmetical features in [8]. Roughly speaking, an important observation by Bourgain is the following: Let (n i ) be a finite set of modes satisfying |n 1 | ≥ |n 2 | ≥ · · · and then unless n 1 = n 2 , one has which follows from the relations and The estimate (1.6) is essential to control the small divisors which arise during the KAM iteration. However, there are many Hamiltonian PDEs do not satisfy (1.6) such as the 1-dimensional nonlinear wave equation. The reason is that (1.8) is no longer true since the linear growth of the frequencies. The quadratic growth of the frequencies (which is also considered as the separation property) is important to overcome the small divisors specially for high-dimensional PDEs. When FNLS with s 0 ∈ (1/2, 1) is considered, the main part C(s 0 )n 2s0 of the frequencies for any n will goes n 2 as s 0 → 1 and tends to |n| as s 0 → 1/2. It is helpful to understand how the separation property is useful to overcome the small divisors. In this paper, the relation (1.8) is replaced by where [C(s 0 )n 2s0 ] represents the integer part of C(s 0 )n 2s0 . Here, we see clearly that the estimate (1.6) fails when s 0 = 1/2. Actually, the main part C(s 0 )n 2s0 of the n-frequency is no longer an integer which is also a problem to define the Diophantine conditions newly. Since C(s 0 )n 2s0 − [C(s 0 )n 2s0 ] ∈ (0, 1) and V n ∈ [−1, 1], the new variable V n (V ) can be considered as new parameters by denoting V n (V ) = V n + C(s 0 )n 2s0 − [C(s 0 )n 2s0 ] for any n, which finally derive the relation (1.9). Obviously, the condition (1.9) is weaker than (1.8). On the other hand, different from Bourgain [8] in which zero-moment condition is satisfied, we need the sub-exponentially decaying of f (n) together with relatively faster (of course slower than Schrodinger equation case) of normal modes to do that. For this purpose, we impose Gevrey smooth condition on the function f (x), that is, one has Thus we can use the properties (1.9) and (1.10) to guarantee |n 1 | + |n 2 | can be controlled by j≥3 |n j | + |n|, which leads to the result of this paper.
To state our result precisely, we will give some definitions firstly.
Let q = (q n ) n∈Z and its complex conjugateq = (q n ) n∈Z . Introduce I n = |q n | 2 and J n = I n − I n (0) as notations but not as new variables, where I n (0) will be considered as the initial data. Then the Hamiltonian R has the form and B akk ′ are the coefficients. Define by supp M akk ′ = {n : 2a n + k n + k ′ n ̸ = 0}, (1.11) and define the momentum of M akk ′ by Moreover, denote by n * 1 = max{|n| : a n + k n + k ′ n ̸ = 0}, Now we define the norm of the Hamiltonian as follows Definition 1.1. For any given ρ > 0, µ > 0 and 0 < θ < 1, define the norm of the Hamiltonian R by  (1) the amplitude of E is restricted as (2) the frequency on E was prescribed to be [C(s 0 )n 2s0 ] + ω n n∈Z ; (3) the invariant torus E is linearly stable. [8] has claimed that the statement held for most (V n ) n∈Z ∈ [−1, 1] Z , but not proven yet.

Some notations and the norm of the Hamiltonian
Given r > 0, let Then we have the following result: Then given any ω = (ω n ) n∈Z satisfying the non-resonant condition (1.15) and for sufficiently small ϵ depending on r, ρ, µ, θ and γ, there exist V * ∈ Π and a real analytic symplectic coordinate transformation Φ : and R 2, * has the form of (2.5) and satisfies

Derivation of homological equations
The proof of Theorem 2.1 employs the rapidly converging iteration scheme of Newton type to deal with small divisor problems introduced by Kolmogorov, involving the infinite sequence of coordinate transformations. At the s-th step of the scheme, a Hamiltonian H s = N s + R s is considered as a small perturbation of some normal form N s . A transformation Φ s is set up so that with another normal form N s+1 and a much smaller perturbation R s+1 . We drop the index s of H s , N s , R s , Φ s and shorten the index s + 1 as +.
We desire to eliminate the terms R 0 , R 1 in (2.2) by the coordinate transformation Φ, which is obtained as the time- and the homological equations become where and The solutions of the homological equations (2.8) are given by and where V n denote the modulated frequencies by readjusting the multiplier (V n ) in (1.1) to ensure at each stage V n = ω n with ω = (ω n ) a fixed frequency.
The new Hamiltonian H + has the form where and (2.14)

The solvability of the homological equations (2.8)
In this subsection, we will estimate the solutions of the homological equations (2.8).
To this end, we define the new norm for the Hamiltonian R of the form as follows: where Moreover, one has the following estimates: where C(θ) is a positive constant depending on θ only.

20)
where i = 0, 1 and C(θ) is a positive constant depending on θ only.
Proof. We distinguish two cases: where the last inequality is based on supp k supp k ′ = ∅. There is no small divisor and (2.20) holds trivially.
In this case, we always assume otherwise there is no small divisor. Firstly, one has n∈Z |k n − k ′ n ||n| where the last inequality is based on Lemma 3.2.

The new perturbation R + and the new normal form N +
Firstly, we will prove two lemmas.

26)
where C(θ) is a positive constant depending on θ only.
Proof. The details of proof had been given in [10] of Lemma 2.4.

in Lemma 2.3. Then for any Hamiltonian function H, we get
where C 1 (θ) is a positive constant depending only on θ.
Proof. The details of proof had been given in [10] of Lemma 2.5.
Recall the new term R + is given by (2.14) and write Following the proof of [9], one has The new normal form N + is given in (2.13). Note that [R 0 ] (in view of (2.9)) is a constant which does not affect the Hamiltonian vector field. Moreover, in view of (2.9), we denote by a00 M a00 will be given in the next section (see (3.26) for the details).
Finally, we give the estimate of the Hamiltonian vector field. Proof. The details of the proof had been given in [10] of Lemma 2.6.

Iteration and Convergence
Now we give the precise set-up of iteration parameters. Let s ≥ 1 be the s-th KAM step.
Finally, we will freeze ω by invoking an inverse function theorem. From (3.31) and the standard inverse function theorem, we can see that the functional equation  (3.33) and by using (3.27), (3.31) implies which verifies (3.10) and completes the proof of the iterative lemma.