KAM TORI FOR QUINTIC NONLINEAR SCHR¨ODINGER EQUATIONS WITH GIVEN POTENTIAL

. This paper is concerned with the 1-dimensional quintic nonlinear Schr¨odinger equations with real valued C ∞ -smooth given potential subject to Dirichlet boundary conditions. By means of normal form theory and an inﬁnite-dimensional Kolmogorov-Arnold-Moser (KAM, for short) theorem, it is proved that the above equation admits a family of elliptic tori where lies small amplitude quasi-periodic solutions with two frequencies of high modes.


Introduction.
In the past few decades, KAM theory has been generalized to the infinite dimensional version and applied to construct the quasi-periodic solutions for some Hamiltonian PDEs. Among those PDEs, the nonlinear Schrödinger equations ( √ −1u t − u xx + V u + f (|u| 2 )u = 0) and the nonlinear wave equations ( u tt − u xx + V u + f (u) = 0) in various situations have been studied by many authors, see [4,5,6,8,10,15,24,25,28,29] for references. As for KAM theory, it is well known that the parameters play an important role in overcoming the small denominator problem. Kuksin [14,12,13] and Wayne [27] firstly considered the potential V as parameters, which are often referred to as the external parameters. Generally speaking, the aforementioned papers conclude that there exist many quasi-periodic solutions for "most" potentials V . However, for a prescribed potential V , there are no external parameters any more, which makes it necessary to extract parameters from the nonlinear term by means of the Birkhoff normal form technique. For the case V (x) ≡ m in the nonlinear Schrödinger equations, Kuksin and Pöschel [15] obtain the parameters from the nonlinear term |u| 2 u while Liang and You [18] get it from the term |u| 4 u. For the nonlinear wave equations, the same result holds true, but there are some restrictions on m, see [10,25] for details. If the potential V is not constant, it becomes very difficult to get a suitable normal form to extract parameter from the term |u| 2 u (or u 3 ). Yuan and Du [29,9] give a positive answer by choosing high modes as tangent frequencies. Besides, we mention that much progress has also been made on the applications of infinite KAM theorem to those kind of PDEs with nonlinearity containing spatial derivative, including KdV equations, Benjamin-Ono equations, derivative nonlinear Schrödinger equations and derivative nonlinear wave equations, the corresponding KAM theorems which are applicable to the unbounded Hamiltonian vector fields are developed to construct the quasi-periodic solutions for these PDEs, see, for instance, [3,1,11,16,17,19,20,21,22,30] for references.
The aim of this paper is to investigate the following nonlinear Schrödinger equations with real valued smooth potential on the finite x-interval [0, π] subject to Dirichlet boundary conditions u(t, 0) = u(t, π) = 0, t ∈ R, and construct its quasi-periodic solutions via KAM theory. We show that this equation possesses lots of small amplitude quasi-periodic solutions lying on 2-dimensional elliptic invariant tori.
Following the idea in [12], we treat equation (1) as an infinite Hamiltonian dynamical system on some suitable phase space D, one may choose the usual Sobolev space, for instance, D = H 1 0 ([0, π]). The Hamiltonian for the nonlinear Schrödinger Equation (1) then reads , ·, · represents the the usual inner-product of L 2 ([0, π]), and equation (1) can be rewritten in the Hamiltonian forṁ where the gradient of the Hamiltonian H is defined with respect to ·, · . The time-quasi-periodic solutions of equation (1) we are going to construct are of small amplitude, a conventional way is to treat the high order term |u| 4 u as a perturbation of the linear PDE √ −1u t = u xx − V (x)u. Let φ µ (x) and λ µ (µ = 1, 2, · · · ) be the basic modes and frequencies for this linear PDE under the Dirichlet boundary conditions (2). It should be noted that when V (x) ≡ m with m a positive real number, then the basic modes and frequencies are trivial, that is, φ µ (x) = 2 π sin µx and λ µ = µ 2 + m. However, when V (x) is a given smooth potential, then the basic modes are not trivial at all. Actually, from [26] we know that φ µ (x) and λ µ admit some asymptotic expressions. To be more specific, the frequencies λ µ could be expanded in the following manner where V 0 = 1 π π 0 V (x)dx,V represents some constant depending on the potential V (x) and O( 1 µ 4 ) denotes the high order term. Hence each solution is the superposition of harmonic oscillations of these modes in the following form where [·] denotes the integral part and Then there exists a set C * ⊂ P 2 with positive Lebesgue measure, a family of Diophantine 2-tori T J [C * ] = ∪ I∈C * T J (I) ⊂ S J over C * , and a Lipshchitz continuous embedding T J [C * ] → D, which is a higher order perturbation of the trivial inclusion Φ 0 : S J → D restricted to T J [C * ], such that, the restriction of Φ to each T J (I) in the family is an embedding of invariant rotational 2-tori for the nonlinear Schrödinger equation (1). In addition, the invariant tori carry plenty of quasi-periodic solutions of high modes.

Remark 1.
Let us make some comments on our main result with previous work. On one hand, Theorem 1.1 in the present paper extends the work of [18], where the authors deal with a special case when V (x) ≡ M, M is a constant. Considering the nonlinear term is of the form |u| 4 u, before applying KAM theorem, one has to ensure the combinations of six frequencies do not vanish, i.e., This condition is easy to verify when V (x) ≡ M , since at this time the indexes of the frequencies satisfy the zero-momentum condition, i.e., i ± j ± k ± l ± m ± n = 0. However, for a general potential V (x) in the present paper, the zero-momentum condition does not hold true anymore, which leads to the key difficulty in this case. We manage to fix it in Lemma 4.1. On the other hand, compared with [9], where the nonlinear term is of the form |u| 2 u, one only needs to check that the combinations of four frequencies do not vanish. Things turn to be more involved in the present paper, we adopt an more effective strategy to solve it, see Lemma 4.1 for details.
Remark 2. The given potential V (x) here is assumed to be real valued, which is necessary to make sure that Eq. (1) possesses the conserved quantity 2π 0 |u| 2 dx. It is worth noting that this conservation law makes the tangential and normal frequencies to be affine function, which makes it easier to check the measure estimate part of the KAM theorem developed by Pöschel in [24].
To specify the large number N , by above Lemma 2.1, we assume that Let then coefficient κ µ in Lemma 2.1 admits the following asymptotic expansion.
Lemma 2.2. We simply have moreover, Proof. Through direct computations, this lemma can be proved. See [29] for details.
Finally, we give an useful lemma, which can be found in the appendix in [8].
, then for any n > 0, there exists a constant C n such that 3. The Hamiltonian. The Hamiltonian for the nonlinear Schrödinger Equation (1) is where We rewrite H as a Hamiltonian in infinitely many coordinates by making the ansatz The coordinates are taken from the Hilbert space a,p of all complex-valued sequences q = (q 1 , q 2 , · · · ) with q 2 a,p = j≥1 j 2p |q j | 2 e 2aj < ∞.

One then gets the Hamiltonian
with where We equip the phase space a,p × a,p with symplectic structure √ −1 2 j≥1 dq j ∧ dq j , then the equations of motion arė They are the classical Hamiltonian equations of motions for the real and imaginary parts of q j = x j + √ −1y j written in complex notation. Rather than discussing the above formal validity, we shall, use the following elementary observation.
The proof can be found in [15]. Next, we study the regularity of the gradient of G in the following lemma.
Lemma 3.2. For p > 1 and a ≥ 0, the gradient Gq is real analytic as a map from some neighbourhood of the origin in a,p into a,p with ||Gq|| a,p = O(||q|| 5 a,p ). Proof. In view of (14), it is clear that Gq j = u 5 , φ j . Then, taking advantage of the Remark 3.3 in [2] and the Lemma 2.3, we have where the second inequality has using the algebraic property of the norm in Sobolev Space.
In the rest of this section, we shall calculate the coefficients G iijjkk , which are useful when checking the conditions of KAM theory.
Proof. On account of (6), one gets that where in the last line we have used the Lemma 3.2 in [29]. Due to (11), the tail Then we continue to calculate the integral and obtain that in which I, II respectively stand for the first term and the second one in the fourth line and Next we shall give some estimates of I, II. Set V(x) = − 1 2 x 0 V (s)ds. Owing to (6), it is clear that In view of Lemma 2.1, one gets that sup Therefore, As to II, one easily obtains that If j = l = n k , k = 1, 2, due to the fact π 0 φ 2 i dx = 1, then where the technique we have used here is analogous to that of deriving (25) and Otherwise j = l, without loss of generality, we assume that j = n 1 , l = n 2 . By (27), similarly, we have that where the estimate of O( 1 n1 ) is the same as that in (29). Combining the relations (21), (23), (25), (28) and (30), the conclusions (18) hold true. When N ≥ 2π N 2 λ + N 2 φ and j = n 1 , l = n 2 , the estimates (22), (24), (26), (29) and (30) lead to (19). 4. Partial Birkhoff normal form. Since the quadratic part of Hamiltonian (15), does not provide any "twist" condition required by KAM theory, we shall use the normal form technique to get the "twisted" integrable terms from the sixth order terms. To get a two dimensional KAM tori, for simplicity, we choose (q n1 , q n2 ) as tangential variables. All the other variables are called normal ones. In this part, the sixth order terms with at most two normal variables will be cancelled, while the other sixth order terms are left since they have no effect on the tori. Then we define the index sets * , * = 0, 1, 2 and 3 in the following way: * is the set of index (i, j, k, l, m, n) such that there exist right components not in {n 1 , n 2 }. 3 is the set of index (i, j, k, l, m, n) such that there exist at least three components not in {n 1 , n 2 }. Define the resonance sets N = {(i, j, k, i, j, k)}. For our convenience, rewrite G =Ḡ +G +Ĝ, wherē (i,j,k,l,m,n)∈ 3 G ijklmn q i q j q kqlqmqn .
To removeG, we need the following lemma.
Due to the assumption {i, j, k} {l, m, n} = ∅, we know that i = j = k = n 2 and l = m = n = n 1 , or vice versa. It contradicts the condition i 2 +j 2 +k 2 = l 2 +m 2 +n 2 .
Without loss of generality, we assume that i = n 1 , n 2 . we just consider the case j = k = n 1 , l = m = n = n 2 . Indeed, if j = k = n 2 , l = m = n = n 1 , then i 2 + 2n 2 2 = 3n 2 1 which is impossible to hold true by the fact that n 2 1 ≤ n 2 . Thus one obtains i 2 + 2n 2 1 = 3n 2 2 . Again, due to the fact that n 2 1 ≤ n 2 , we know i > n 2 . Then it is easy to get In this subcase, without loss of generality, we either have (a) k = n 1 , l = m = n = n 2 , and i, j = n 1 , n 2 ; or (b) j = k = n 1 , m = n = n 2 , and i, l = n 1 , n 2 .
Next we transform the Hamiltonian (15) into the partial Birkhoff form of order six so that the KAM Theorem can be applied. Proposition 1. For any given n 1 , n 2 satisfying (31), there exists a real analytic, symplectic change of coordinates Γ from some neighbourhood of the origin in a,p into a,p that takes it into where XḠ, XĜ and X K are real analytic vector fields defined in a neighborhood of the origin in a,p , and taking values in a,p with and |K| = O( q 10 a,p ). The proof of this proposition can be derived directly from [2], we omit it.

5.
Proof of the main theorem. In this section, we shall prove the main theorem, our proof can be divided into the following five steps.
Step 2. Check the non-degeneracy condition of frequencies.
Case I: |l| = 1. In this case, we discuss it from several possibilities.
There are four similar subcases under this situation, we only deal with the case when j = 2n 1 .
, which yields that k 1 ∈ Z, k 2 ∈ Z, this cannot happen. Case V: |l| = 2, and l i = l j = −1, i = j. It is similar with the Case III, we omit it. Similarly, we mention that, if we assume that k, α + l, β = 0 holds true for some k ∈ Z 2 and 1 ≤ |l| ≤ 2, then one can obtain F k + B T l = 0. This completes the proof of this lemma.
Step4: Checking the regularity property: To this end, it requires to define the phase space and some norms. Let P a,p = T 2 C × C 2 × l a,p × l a,p (x, y, z,z) represent the phase space, in which T 2 C denotes the complexification of the 2-torus T 2 . Define D(s, r) = {(x, y, z,z) ∈ P a,p : | x| < s, |y| < r 2 , z a,p + z a,p < r}, where | · | represents the usual sup-norm for complex vector and · a,p denotes the norm defined in the Hilbert space l a,p . One can define the weighted norm in the following form |U | s,r = |x| + 1 r 2 |y| + 1 r z a,p + 1 r z a,p for U = (x, y, z,z) ∈ P a,p . Let Π denote the parameter set, i.e., ξ = (ξ 1 , ξ 2 ) ∈ Π. For a mapping W : D(s, r) × Π → P a,p , set its Lipschitz semi-norm as follows: where the supremum is taken over the parameter set Π. Let X R be the Hamiltonian vector field with respect to the symplectic structure dx ∧ dy + √ −1dz ∧ dz, i.e., X R = (∂ y R, −∂ x R, ∇ z R, ∇zR).

From
Step 3, one has Ω j (ξ) = j 2 + · · · + j −1 , where the dots stand for fixed lower order term in j. To be specific, there exists a fixed, parameter-dependent sequence