Here the coordinate (q i , p i ) is indexed to the frame of reference of the system, p i is generalized momentum associated with the generalized coordinate q i . In Newtonian mechanics, the time evolution of both position and velocity are computed by applying Newton's second law to the total force being exerted on each particle in the system. In contrast, the time evolution state in Hamiltonian mechanics is obtained by computing the Hamilto-Here j = √ 1 + j 2 . The questions now become : once the well posedness (local or global) is proved, assume that the initial data is sufficiently small, we want to know -Do invariant tori persist or the perturbation kill these tori ?

This fact is a powerful tool in studying Hamiltonian equations. Indeed, KAM theory and Birkhoff normal form theory are based on the search for symplectic transformations in order to conjugate the original Hamiltonian to a normal form. These symplectic transformations are founded by solving homological equations.

KAM theory

Over the past half century, the KAM theory has played an important role in studying long time behavior of solutions of non integrable Hamiltonian system. The integrable Hamiltonian system admits many invariant tori in phase space. Different initial conditions form different invariant tori. The KAM theorem states that most of such tori persist under small Hamiltonian perturbations. This result is valid under certain conditions of suitable regularity and sufficiently irrational frequencies. The persistence of such invariant tori implies that the motion continues to be quasiperiodic. The name KAM theory comes from Kolmogorov, Arnol'd and Moser who initiated the theory.

We talk about the history of the KAM theory. In 17 century, Kepler announced his study describing the orbit of a planet around its Sun as an ellipse. However, with Newton'law of gravitation, physicians and mathematicians then realized that the disturbance due to interaction between planets makes their orbits more complicated than their in Kepler'law. Scientists want to know how much disturbance affects the trajectories of the planets. Started by Poincaré, mathematicians believed that a small disturbance in a long time could push the planets far away from Kepler's orbits and the stable trajectories are exceptional. Indeed, considering general Hamiltonian tori, it was a common belief that an integrable system can be turned into an ergodic one on each energy surface under

Birkhoff normal form

KAM theory is no doubt a very useful tool in studying Hamiltonian PDEs. However, in order to prove a KAM result, we usually need to assume some undesirable hypotheses, which are not always satisfied. Mathematicians want to know more about the behavior of the solutions lying outside KAM tori. The idea of Birkhoff normal form theory came up and soon played a crucial role in studying Hamiltonian system. Unlike the KAM procedure just works for a set of finite, nondegenerate invariant tori for every time, the Birkhoff normal form works for not any special tori but for all tori in a finite but sufficiently long time. Birkhoff normal form was first derived by Birkhoff and Lewis [BL34] in 1934

). In all these results, the integrability of the unperturbed linear system plays a crucial role in controlling the perturbed spectrum.

In high dimensional cases, there is still a serious problem of perturbations. Typically, 2 2m ∆ + V (x) forms quantum states with discrete state energy. The state energy E is proportional to angular frequency ω. Schrödinger applied his equation to the hydrogen atom. The square of the wave function Ψ 2 gives the probability of finding the electron at position x and time t.

By replacing the energy E in Schrödinger's equation with a time-derivative operator, Schrödinger then generalized his wave equation to describe how a system changes from .

(1.1.3)

Here the independent variable x corresponds to distance along the channel and t is proportional to elapsed time. The quantity η = η(x, t) corresponds to the depth of the water

Acknowledgement

Foremost, I owe my deepest gratitude to my supervisor Benoît Grébert. Thank you for giving me the opportunity to do this research and giving me invaluable guidance throughout this research. I could not have done the thesis without your support. Your comments, corrections, and suggestions to reading my works have kept my work on track from the beginning of my master to the final draft of my thesis. I would also like to thank you for your patience to answer my questions each time I came to your office. Your knowledge, dynamism, enthusiasm, and sincerity have deeply inspired me. It was a great privilege and honor to work and study under your guidance.

I am expressing my thank to Roberto Feola, Dario Bambusi, Joackim Bernier, Cristina Benea. It was my luck to take part in the conferences in Pisa and Mexico with you.

I am grateful to Professor Le Minh Ha, who encouraged me to do my master in Nantes.

I would like to show my gratitude to the team of Secretaries : Anaïs, Stéphanie, Annick, Béatrice and Brigitte. Administrative procedures are too confusing and complicated for me without your conscientious guide. Thanks to two computer experts Eric and Saïd for your timely assistance. Thanks to two librarians Anh and Claude for spending your time to help me find books. I would also like to thank Éric Paturel, director of the department. I am sincerely grateful to the Centre Henri Lebesgue and for funding my master 2 and my thesis. I deeply thank the entire Jean Leray laboratory for the quality of its welcome which makes years spent here a great time. I show gratitude to my committee Erwan Faou and Nicoleau François. Special thanks to my rapporteurs Pietro Baldi and Philippe Bolle for spending time to read my thesis and giving me correct comments and suggestions. I thank also to team of juries Valeria Banica, Joackim Bernier, Erwan Faou and Laurent Thomann.

I am indebted to many of my colleagues to support me throughout my thesis. I am grateful to Germain, Caroline, Matilde, Matthieu, Solène, Côme, Fabien, Maha, Thomas, Hala, Zeinab, Claire, Alexandre, Mael, Antoine, Mohamad, Hèlène, Ngoc Anh and Meissa four your friendship. My four years in Nantes would have been boring without you. Meissa, it is my pleasure to share the office with you. I will never forget the time playing wooden puzzle game with you. Caroline, thank you for inviting me to visit your house, I still keep the video when we played seesaw on the field. Fabien, Thomas and Mohamad, climbing wall with you is one of my best memories in Nantes. Fabien, you are a very kind friend. Many times, I am proud to say " I have a friend, his name is Fabien". Mohamad, each time you came up with an idea, you knocked my office and wrote down on the blackboard quickly everything in your mind. It's happy to be believed although I did not help you much. Solène, thank you for your advice about climbing skills, although until now I am still very bad at it. Claire, I have to confess that I like watching you smile and say " hey, Trung". Germain, you have tried many times to teach me French, I am sorry that I have not improved much. Your smile is very beautiful too. Maha, Hala and Zeinab, you gave me an impression that Lebanese girls are all friendly, humorous, and of course beautiful. I hope that I will have a chance to visit your country. Finally, I would like to give my sincerely thank to Ngoc Anh, you are my sister. You helped me a lot during my four years in Nantes. Words are powerless to express my gratitude.

My thanks go to my friends : Duy Linh, Thuy Duong, Quoc Tuan, Tu Anh, Tam, Binh, Lan Anh, Thanh Nguyen, Duc, Dong Nhi, Phuong Thuy, Giang, Nam, Khanh, Chau, Hien. I especially express my gratitude to Duy Linh for sharing your room with me, teaching me how to cook and many, many other things. Thanks Quoc Tuan for many times helping me with latex problems. My first paper could have been not completed without accurate computations from Lan Anh and Duc. Thanh Nguyen, thank you for your encouragement, I hope that one day I will have an opportunity to visit Korean.

Finally, I am extremely grateful to my parents and my brother for their love, caring and sacrifices for educating and supporting me for my future. I am very much thankful to my girlfriend for her love, understanding and continuing support to complete this research. Every time I was ready to quit, you encouraged me and I am forever grateful.

Table of Contents

Introduction

General introduction 1.Partial differential equations

In the past decades, partial differential equations have been intensively studied. PDEs play an important role in scientific fields, especially in physics and engineering. They arise from many physical considerations, like fluid dynamics, quantum mechanics, statistical mechanics, heat diffusion, N-body problem. These different physical phenomena give rise to various mathematical models and establishes a strong connection between Mathematics and Physics.

Without being exhausted, we mention here some interesting examples. One of the most well-known PDEs is the Schrödinger equation, which describes the probability density of the presence of a non-relativistic massive particle and named after Erwin Schrödinger, who first introduced the equation in 1926 to study electrons in the atom. The discovery of the Schrödinger equation was a significant landmark in the study of quantum mechanics. To describe relativistic massive particles, we have the Dirac equation, which was derived by British physicist Paul Dirac in 1928. In the same context, we have the Klein-Gordon equation (1926).A year after the publication of the Schrödinger equation, Hartree published his research, what is now known as the Hartree equation, considering the electron systems in a spherical potential. Considering fluid dynamics, we have the Korteweg-De Vries (KdV) equation ( 1895), Boussinesq equation (1872), Navier Stokes equation (1845), and also the equation of Burgers (1948), which is a mathematical model of waves on shallow water surfaces. The KdV equation was first introduced by Boussinesq in 1877 and rediscovered by Diederik Korteweg and Gustav de Vries (1895). Another example for fluid dynamics is the Boussinesq equation, which is named after Joseph [START_REF] Boussinesq | Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond[END_REF], who first derived it to study solitary waves. The Navier-Stokes equations describe Introduction the motion of fluid relating pressure, temperature, and density. The Burgers' equations appear in various areas of applied mathematics, such as gas dynamics, traffic low and nonlinear acoustics. Considering the heat diffusion phenomenon, Joseph Fourier introduced the theory of the heat equation in 1807. The heat equation, along with its variants, also appears in many fields of applied mathematics, like probability theory and financial mathematics.

All of these equations indicate the importance of PDEs. Solving these equations helps us understand physical phenomena. Studying these equations, we deal with many questions, like questions about the existence of solutions, the uniqueness of solutions, the behavior of solutions, how long the solutions exist and how we can approach the solutions,... Unfortunately, there are many different types of PDEs and there is no general method or general theory for PDEs. Different methods have been developed to deal with many of the individual equations. Fortunately, PDEs considering conservative physical phenomena are Hamiltonian, which corresponds to the total energy of the system, both kinetic and potential energy.

In this thesis, we focus on study Schrödinger equations, which describe the wave function of a quantum mechanical system, and Boussinesq system, which was derived to study the water way in a shallow water regime. Both systems can be written in Hamiltonian form.

Hamiltonian

In 1833, starting from Lagrangian mechanics, William Rowan Hamilton developed a reformulation of Newtonian mechanics, known as Hamiltonian mechanics, which then historically played an important role in the development of quantum physics.

In classical mechanics, the time evolution of a physical system is obtained by Hamiltonian equations

dp i dt = - ∂H ∂q i , dq i dt = ∂H ∂p i for 1 ≤ i ≤ N.
Introduction nian of the system. The system with many degrees of freedom allows the exchange of energy between different modes, which makes its time evolution complicated and may cause chaotic behavior.

Hamiltonian mechanics is equivalent to Lagrangian mechanics under Legendre transform when holding q and t fixed and defining p as the dual variable. However, Hamiltonian mechanics along with its symplectic structure has demonstrated its strength in studying physical systems. In classical mechanics, the Hamiltonian induces a symplectic structure on smooth functions and forms a symplectic manifold, which is called the phase space. The Hamiltonian then can be generalized to quantum mechanics.

The Hamiltonian of a closed mechanical system is commonly expressed as the sum of functions corresponding to the kinetic and potential energies of a system in the form H = H 0 + P where H 0 is the kinetic energy and corresponds to the linear part of the dynamic system while P is the potential energy and normally corresponds to nonlinear part. Generally, the Hamiltonian generates the time evolution of solutions

iu t = X H (u) = Au + f (u)
where X H is the Hamiltonian vector field, A is a linear operator (Laplace operator in Schrödinger equation) diagonalized on an orthonormal basis of the phase space (φ j ) j∈J , f (u) is a nonlinear term, and u(x, t) is a complex function. The linear equation gives us solutions in an explicit form u(t, x) = j∈J c j e iω j t φ j (x) where frequency ω j denotes the eigenvalue of A associated with the eigenfunction φ j . We see that different linear modes form different trajectories without interacting each others. These linear trajectories define finite or infinite invariant tori in the phase space.

We define the Sobolev norm and Sobolev space H s H s := {u(x) = j c j φ j (x) | u 2 s := j∈J c 2 j j 2s < ∞}.

Introduction

-Whether the solutions remain bounded for all time or if there exist initial conditions that give unbounded solutions ? -Even when we do not know for all time, can we answer for a very long but finite time ?

Much effort has been done to answer these questions, leading to the birth of many methods. KAM theory and Birkhoff normal form theory are two of the most well-known methods.

Poisson bracket

The Poisson bracket, which is named in honor of French mathematician, engineer, and physicist Siméon Denis Poisson (1782-1840), plays a central role in Hamiltonian dynamical equations. In canonical coordinates (q i , p i )(q i , p i ), given two functions f (p i , q i , t), g(p i , q i , t) the Poisson bracket is defined in the form

{f, g} := N i=1 ( ∂f ∂q i ∂g ∂p i - ∂f ∂p i ∂g ∂q i ).
The Poisson brackets of the canonical coordinates, which are called the fundamental Poisson brackets, are {q i , q i } = 0 {p i , p i } = 0 {q i , p j } = -{p j , q i } = δ i,j here δ ij is the Kronecker symbol.

Generally, let M be a smooth manifold and ω be a closed non degenerate differential 2-form on M, then the pair (M, ω) is a symplectic manifold and ω is a symplectic form. Let f and g be two differentiable functions depending on the phase space and time, their Poisson bracket {f, g} is given as {f, g} = ω(X f , X g ) = X g f where X f denotes the vector field generated by f and X g f denotes the vector field X g applied to the function f as a directional derivative. The Poisson bracket of two differentiable functions is a differentiable function in the phase space. The algebra of smooth functions in the phase space together with the Poisson bracket form a Poisson algebra, which is a Lie algebra under the Poisson bracket. Every symplectic manifold is Poisson where H is the Hamiltonian and u is a complex function. The Poisson bracket in (1.1.1) is defined as

{f, g} = -i(∂ u f ∂ u g -∂ u g∂ u f )
A Hamiltonian dynamical system typically has constants of motion besides the energy, these constants of motion commute with the Hamiltonian under the Poisson bracket. If f, g are two constants of the motion, then their Poisson bracket {f, g} is also a constant of the motion. A Hamiltonian system that contains a maximal set of conserved quantities, i.e., there exists a maximal set of constants of motion, is completely integrable.

The Poisson bracket is preserved under symplectic transformations ( canonical transformations preserve the bi-linear form ω). To be more specific, given two functions f, g This property is especially useful in studying Hamiltonian mechanics. Let Φ S be a symplectic transformation and H be a Hamiltonian, then

H • Φ S = H + {S, H} + n≥2 1 n! ad n S (H)
where ad 1 S (H) := {S, H}, ad n S (H) = {S, ad n-1 S (H)} ∀n ≥ 2.

Introduction an arbitrarily small perturbation. This means that for a long time the system forgets its initial state and the trajectories eventually visit almost all points in any subset of the phase space.

However, in 1954, Kolmogorov [START_REF] Nikolaevich | On conservation of conditionally periodic motions for a small change in Hamilton's function[END_REF] in his talk at the International Congress of Mathematicians in Amsterdam, announced that the majority of tori survive and instability is possible but very rare. Arnold [START_REF] Vi | Proof of a theorem of AN Kolmogorov on the conservation of quasiperiodic motions under a small change of the Hamiltonian function[END_REF] and Moser [START_REF] Möser | On invariant curves of area-preserving mappings of an annulus[END_REF] then completed his proof. The general result is known as the KAM theorem.

Mathematicians have made significant advances in the KAM theory since then. The theory was originally applied to perturbed Hamiltonian PDEs in one dimension. In [START_REF] Sergei B Kuksin | Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum[END_REF][START_REF] Kuksin | Nearly integrable infinite-dimensional Hamiltonian systems, sous la dir[END_REF], Kuksin used KAM methods to prove the existence of quasi periodic solutions of nearly integrable Hamiltonian equation in infinite dimension. In [START_REF] Wayne | Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory[END_REF], Wayne proved the existence of periodic and quasi periodic solutions for nonlinear wave equations in dimension one with Dirichlet boundary condition. The theory was then applied to the nonlinear wave equation with periodic boundary conditions [START_REF] Chierchia | KAM Tori for 1D Nonlinear Wave Equations with Periodic Boundary Conditions[END_REF], Klein-Gordon equation [START_REF] Alexander | The nonlinear Klein-Gordon equation on an interval as a perturbed sine-Gordon equation[END_REF] and nonlinear Schrödinger equation [START_REF] Kuksin | Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrodinger equation[END_REF]. While others need external parameters to verify the non resonant condition, the result in [START_REF] Kuksin | Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrodinger equation[END_REF] is remarkable since it is the first result for a Hamiltonian without an external parameter. All of these results are for Hamiltonian equations in one dimensional context.

The study of KAM theory in multidimensional space has just started recently (see [START_REF] Bourgain | Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations[END_REF][START_REF] Bourgain | Green's function estimates for lattice Schrödinger operators and applications[END_REF][START_REF] Håkan | On reducibility of Schrödinger equations with quasiperiodic in time potentials[END_REF][START_REF] Hakan | KAM for the nonlinear Schrödinger equation[END_REF]). In these just mentioned papers, the authors considered Hamiltonian PDEs with external parameters in the linear part, by which a non-resonant condition is achieved. Usually, the parameter enters the equation through the potential term V (x)u(t, x) or V (x) * u(x), where the potential V depends on the parameter. The techniques developed in [EK09 ; EK10] then has been extended in [START_REF] L Hakan Eliasson | KAM for the nonlinear beam equation[END_REF] to a KAM result without external parameters (see also [START_REF] Procesi | A KAM algorithm for the resonant non-linear Schrödinger equation[END_REF]). The approach in [EK09 ; EK10] allows to analyse the linear stability of the KAM tori. We remind that a solution of a nonlinear equation is called linear stable if the linearization of the equation at this solution has linear operator whose spectrum contains only pure imaginary eigenvalues. In [START_REF] Procesi | A KAM algorithm for the resonant non-linear Schrödinger equation[END_REF] (see also [START_REF] Procesi | A KAM algorithm for the resonant non-linear Schrödinger equation[END_REF][START_REF] Procesi | The energy graph of the non-linear Schrödinger equation[END_REF][START_REF] Wang | Energy supercritical nonlinear Schrödinger equations : quasiperiodic solutions[END_REF]), applying a KAM algorithm, the authors proved the existence of large families of stable and unstable quasi periodic solutions for the NLS in any number of independent frequencies. The considered quasi periodic solutions base on non-degenerate sets A of linear modes.

Proving the KAM theory involves verifying the nonresonant condition of the frequencies. The nonresonant condition becomes increasingly difficult to verify for systems with Introduction more degrees of freedom. In many cases, the frequencies by themselves are resonant and we must use external parameters to achieve the nonresonant condition.

We are then interested in the stability and instability of the KAM tori. Although, using KAM theory, the stability is frequently observed, the instability also occurs. Such invariant tori exhibit hyperbolic directions which induce instability of the tori (see [START_REF] L Hakan Eliasson | KAM for the nonlinear beam equation[END_REF]).

We consider the Hamiltonian perturbation H = h 0 + f, with the small perturbation f. Let us denote (r, θ) ∈ R n × T n the action-angle variables associated with invariant tori, and z = (ζ, η), η = ζ the external modes. For any perturbation f , we define its jet function f T , the effective part of f, as a quadratic truncation of Taylor expansion of f around the origin r = 0, z = 0 :

f T = f (0, θ, 0) + ∂ r f (0, θ, 0) • r + ∂ z f (0, θ, 0) • z + 1 2 ∂ 2 z f (0, θ, 0)z, z .
In KAM theory, we are looking for an an analytic, near-identity, symplectic transformation Ψ, which puts the original Hamiltonian H = h 0 + f to a desirable form

(h 0 + f ) • Ψ = h + g
where h is a Hamiltonian on normal form and the perturbation g has its effective part gT vanishing.

The idea of KAM theory is to do an iterative procedure. More precisely, considering the original Hamiltonian H = h 0 + f, we search for an analytic, near-identity, symplectic transformation Φ S = Φ t S | t=1 that puts H into a new form

H + := H • Φ S = h + + f + where -h + is a Hamiltonian, close to h 0 , i.e, |h + -h 0 | ∼ O(ε).
In addition, h + commutes with the linear Hamiltonian h 0 , i.e., {h 0 , h + } = 0.

f + is a new perturbation, whose jet function f T + smaller than f T . Indeed, assuming that the initial data is sufficiently small, we are finding S such that f T + ∼ (f T ) α with α > 1 (α = 3 2 for example). We say that the Hamiltonian h + is in normal form. Since it commutes with h 0 , the normal Hamiltonian does not affect the trajectory of the linear Hamiltonian equation. Once S is achieved, we iterate this procedure to obtain a sequence of symplectic transformations Introduction Φ S j such that

H j+1 := H j • Φ S j = h j + f j
Of course, we want this procedure to converge. The convergence problem relates to the appearance of small divisors. Indeed, in each step of the KAM procedure, we have to solve homological equations

f T j + {h j , S j } = h + + O(ε α ).
The Hamiltonian h + is in normal form. Putting this equation in Fourier formula leads us

to estimate divisors ω • k, ω • k ± Λ s , ω • k ± Λ s ± Λ t , with 0 = k ∈ Z n
, where the eigenvalue Λ s corresponds to external modes. Our procedure would be false if these divisors are too small. In the KAM procedure, one usually wants to find a control of these small divisors, likely the Diophantine inequality

| ω • k | ≥ α |k| τ , | ω • k ± Λ s | ≥ α |k| τ , | ω • k ± Λ s ± Λ t | ≥ α |k| τ ,
for all 0 = k ∈ Z n and some fixed constants α and τ. The major difficulty is that the frequency ω moves during the KAM iteration. The nonlinearity f T j decreases super exponentially of size ε jα (α > 1) allow to the convergence of symplectic transformations

Φ S 1 • Φ S 2 • • • • → Φ ∞ and the normal form h j → h ∞ .
Since Φ ∞ is close to identity, one can talk about the existence of periodic or quasi periodic solutions of the original equation.

Introduction

(see also [START_REF] Lewis | Sulle oscillazioni periodiche di un sistema dinamico[END_REF][START_REF] Moser | Proof of a generalized from of a fixed point theorem due to gd birkhoff[END_REF]) where they consider the neighborhood of elliptic, non-constant, periodic orbits of Hamiltonian systems. In their paper, they put the Hamiltonian systems in fourth order normal form, namely

H = H 0 + G 4 + R 5
where G 4 is a homogeneous polynomial of degree 4 depending only on the actions and R 5 is a remainder term having a zero of at least fifth order at the origin.

The idea of Birkhoff normal form theory was then intensively studied in [Bam03 ; BG06 ; BDGS07 ; GIP09 ; Del12 ; FGL13 ; FI20 ; BG20]. Birkhoff normal form theory describes the behavior of solutions in neighborhoods of elliptic equilibrium points. Typically, it ensures some properties of stability for a finite but very long time.

In the finite dimensional context, we consider the Hamiltonian H = H 0 + P where P is a smooth function having a zero of order at least 3 at the origin, H 0 is the linear Hamiltonian

H 0 = j∈J ω j |u j | 2 .
Provided nonresonant hypothesis, the Birkhoff normal form theory states that, for each r ≥ 1, one can conjugate H into a normal form

H • T r = H 0 + Z + R r (1.1.2)
where -T r is a real analytic symplectic transformation, -the polynomial Z is of order r + 2 depending only on the actions I j := |u j | 2 , -the remainder term R r has a zero of order r + 3.

Assuming the initial data is of size ε 1, as a consequence of Birkhoff normal form, the solution remains bounded in the ball of radius 2ε with center at the origin for times of order ε -r . Moreover, the solutions remain close to a torus of maximal dimension at a distance of size ε r 1 up to times of order ε r 2 , with r 1 + r 2 = r + 1. In [Bam03 ; BG06], the authors generalized Birkhoff normal form theory to infinite dimensional Hamiltonian systems. In these papers, the authors applied to nonlinear wave equations and nonlinear Schrödinger equations to obtain long time existence and bounds of solutions. The point here is that Introduction instead of considering the whole extension of PDEs, we split the phase variables in two groups : low modes and high modes. Precisely, we fix a positive integer number N then write u = u + u with u = |j|≤N u j e ijx and u = |j|>N u j e ijx . Then one observes that monomials with more than two high mode variables in their expression are not relevant, since their vector field is already small. This was proved in [START_REF] Bambusi | Birkhoff normal form for partial differential equations with tame modulus[END_REF] using tame inequality, namely

uv H s ≤ C s ( u H s v H 1 + u H 1 v H s ). Note that u H s ≤ u H s N s-1
for s ≥ 1. This term is negligible when N is sufficiently large. So that if the nonlinearity satisfies a tame modulus condition, it is always possible to put H in the form (1.1.2), where the remainder term R r is of order r + 5/2 and Z is a polynomial of degree r + 2 containing only monomials which are "almost resonant". Assuming also the nonresonant condition, then H can be put in integrable Birkhoff normal form, i.e., Z depends only on the actions.

As in the KAM procedure, the proof of Birkhoff normal form theory relates to solve homological equations, likely to solve

P + {H 0 , S} = Z + + O(ε α )
where P is a homogeneous polynomial of order at least three, Z + depends only on the actions. Expanding this equation leads us to prove a nonresonant condition for small divisors ω • k with 0 = k ∈ Z ∞ . However, unlike the KAM procedure, the frequency ω does not change during the Birkhoff procedure.

Since we only consider small divisors related to the actions of low modes and maximum two high modes, the nonresonant conditions (at least in one dimension) are usually satisfied. Let's see an example, the nonlinear wave equation (NLW)

u tt -∆u + V (x)u = g(x, u) x ∈ T where V is a C ∞ ,
|ω k 1 + • • • + ω kp -ω 1 -• • • -ω q | ≥ γ µ 3 (k, ) (*)
where

k = (k 1 , • • • , k p ) ∈ (Z d ) p , = ( 1 , • • • , q ) ∈ (Z d ) q with p+q ≤ r, and {|k 1 |, • • • , |k p |} = {| 1 |, • • • , | q |} ; µ 3 (k, )

Reducibility

Another topic of discussion among mathematicians is the existence of quasiperiodic solutions for time-forced PDEs. Typically, a time forced nonlinear PDE can be formulated as a fixed point problem, which can be solved via the Newton algorithm. We linearize the equation around an approximate quasi periodic solution u 0 , then solve this linear equation to obtain u 1 . We continue to linearize the equation around u 1 to get u 2 and then iterate this procedure. Solving the linear equations leads us to considering a time dependent linear operator and its inverse. The idea of reducibility is to conjugate such time dependent linear operator to a time independent diagonal operator.

We first see how this idea works in the context of ordinary differential equations. One considers a linear system of differential equations with periodic coefficients

ẋ = A(t)x t ∈ R; x ∈ R n
where A(t) is an n × n periodic with period T , piecewise continuous matrix. Let Ψ(t) be a fundamental matrix solution of this differential equation. Then

Ψ(t + T ) = Φ(t)Ψ(0) -1 Ψ(T ) ∀t ∈ R.
Moreover, Gaston Floquet(1833) said that there exists a periodic matrix function P (t) and a constant coefficient matrix B such that

Ψ(t) = P (t)e tB ∀t ∈ R.
This mapping gives rise to a transformation, which puts the original equation into an

Introduction

autonomous form

y = P (t) -1 x, ẏ = By.
Then, the reducibility problem for forced PDEs, which is more difficult, has been intensively studied. The KAM theory plays a crucial role in proving reducibility results. In one dimensional context, we quote [Kuk93 ; BG01 ; LY10 ; GT11]. In these papers, the authors adapted a KAM procedure to prove the reducibility of Schrödinger equation with time dependent perturbation. All these results are for equations with bounded potential.

To consider Hamiltonian with unbounded perturbation, a breakthrough strategy has been developed in [START_REF] Baldi | KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation[END_REF][START_REF] Baldi | KAM for autonomous quasi-linear perturbations of KdV[END_REF]. The idea is to apply pseudo-differential calculus to reduce the order of the perturbation. Pseudo-differential calculus allows us to reduce the perturbation to an arbitrary smoothing operator. More precisely, let f and g two pseudodifferential operators of order a and b, respectively, then their commutator [f, g] := f g-gf is again a pseudo-differential operator but of order a + b -1, while considering f g and gf separately is of order a + b. Apply this to a Hamiltonian with pseudo-differential perturbation H = h + P, we regularize it by an analytic, symplectic transformation

H • Φ S = h + P + [h, S] + [P, S] + 1 2 [[h, S], S] + 1 2 [[P, S], S] + • • •
Thanks to the properties of pseudo-differential calculus, we can gain one regularity in each step of regularization after we solve the homological equation

P + [h, S] + [P, S] = OP a+b-1 .
The remainder of the new perturbation's terms are even more regular. Iterating this procedure, we can put H in a form with an arbitrary smoothing perturbation. Then a reducibility scheme is obtained by applying KAM theory. The idea of using pseudodifferential calculus has been demonstrated to be extremely useful in one dimensional context (see [BBM16 ; FP14 ; BM20 ; Mon19 ; Bam17 ; FGP18].

Introduction

we associated the perturbation with a matrix operator which leads to, in KAM procedure, the homological equation be solved blockwise. However, the increasing size of the blocks may cause loss of regularity. This obstacle was overcome differently in [START_REF] Håkan | On reducibility of Schrödinger equations with quasiperiodic in time potentials[END_REF], using geometric arguments, and in [START_REF] Grébert | On reducibility of quantum harmonic oscillator on R d with quasiperiodic in time potential[END_REF][START_REF] Feola | Reducibility of Schrödinger equation on the sphere[END_REF], where the authors used another argument.

The Schrödinger equation

The Schrödinger equation, which was named after Erwin Schrödinger in 1925, is a partial differential equation which is essential in quantum mechanics. The Schrödinger equation describes probability waves of a quantum mechanical system. It gives the evolution over time of the wave functions. The wave functions contain physical information of the system, such as position, momentum, energy, velocity or other physical properties. The equation is used extensively in atomic, nuclear, solid-state, and many other physical problems.

The Schrödinger equation in quantum mechanics is a counterpart of Newton's second law in classical mechanics. Considering a particle of mass m e , its total energy E is sum of the potential energy V (x) at position x and the kinetic energy

p 2 2me p 2 2m e + V (x) = E.
Since energy is conserved, the particle is assumed to be confined to a certain region in space. By replacing p in the above energy equation with a differential operator and using de Broglie relation, Schrödinger showed that the wave function follows a time-independent partial differential equation

EΨ(x) = - 2 2m ∆ + V (x) Ψ(x) .
Here ∆ is Laplace operator. Eigenvalues associated with eigenfunctions of the linear operator -

Introduction one state to another i ∂ ∂t Ψ(x, t) = - 2 2m ∆ + V (x, t) Ψ(x, t) .
Here, the constant i is the imaginary unit, and is the reduced Planck constant. The time dependent Schrödinger equation allows us to calculate the probability of a transition between one atomic stationary state and some other state.

Nonlinear Schrödinger equation

The nonlinear Schrödinger (NLS) equation is a nonlinear variation of the linear Schrödinger equation, applicable to both classical mechanics and quantum mechanics. The equation appears in many areas of physics and are analyzed mathematically by mathematicians. It was derived in the studies of the propagation of light in nonlinear optical fibers and planar waveguides, the Langmuir waves in hot plasma, the small-amplitude gravity waves on the surface of deep inviscid water, magnetic spin waves, and many others. Typically, most weakly nonlinear, dispersive, energy-preserving systems appropriately give rise to the NLS equation. Historically, the NLS equations were first derived by Ginzburg and Landau [START_REF] Vitaly | On the theory of superconductivity[END_REF] in their study of the macroscopic theory of superconductivity and became well-known, especially in connection with the phenomenon of self focusing, after the work about optical beams of Chiao et al [START_REF] Raymond Y Chiao | Self-trapping of optical beams[END_REF]. From then until now, the NLS equation has been intensively studied in many areas of physics and mathematics. In the optical context, Hasegawa and Tappert [START_REF] Hasegawa | Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion[END_REF] first derived the NLS equation taking into account both group velocity dispersion and fiber nonlinearity due to the so-called Kerr effect. Eisenberg et al [START_REF] Hs Eisenberg | Discrete spatial optical solitons in waveguide arrays[END_REF] drew a discrete NLS model for a coupled optical waveguides. The NLS equations on a lattice background were first studied by Efremidis et al [START_REF] Nikolaos K Efremidis | Two-dimensional optical lattice solitons[END_REF]. The experimental realization of Bose-Einstein condensates generates an NLS equation with external potentials. The NLS equation for small-amplitude water waves was derived by Zakharov [Z68] for the case of infinite depth, then Benney and Roskes [START_REF] Dj Benney | Wave instabilities[END_REF] for the case of finite depth.

In one dimensional context, the cubic NLS equation is integrable, which was solved by Zakharov and Shabat [START_REF] Shabat | Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media[END_REF] via the Inverse Scattering Transform -which is a nonlinear Fourier Transform. It admits an infinite number of conserved quantities and multisoliton solutions. For the equation with nonlinearity of higher order or in higher dimensional context, it is not integrable, the phenomenon of wave collapse and turbulence can take place.

Different nonlinear terms form different nonlinear Schrödinger equations, among them

Introduction the cubic NLS and the quintic NLS are most studied. The cubic NLS takes the form i∂ t ψ = -∆ψ + κ|ψ| 2 ψ here ∂ t ψ is a complex vector field. The equation associates with the Hamiltonian

H = dx |∂ x ψ| 2 + κ 2 |ψ| 4 .
The equation is called focusing NLS when κ is negative. The focusing NLS has bright soliton solutions (localized in space, and having spatial attenuation towards infinity). This case was solved by use of the inverse scattering transform, see [START_REF] Shabat | Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media[END_REF]. The equation is called defocusing when κ is positive. The defocusing NLS has dark soliton solutions (having constant amplitude at infinity, and a local spatial dip in amplitude) . See also [GK14 ; GK02 ; KL12 ; Bou99]. In optics, the NLS equation describes the propagation of the wave in fiber optics through a nonlinear medium while for water waves, the solution ψ is related to the amplitude and phase of the water waves. The value of the nonlinearity parameter κ depends on the relative water depth. The NLS equation is focusing on shallow water, with the water depth small compared to the wave length of the water waves, and defocusing on deep water .

Considering purely self-focusing cubic nonlinearity in 2 spatial dimension, it was proved by Vlasov et al [START_REF] Sn | Theory of periodic selffocusing of light beams[END_REF] , the phenomenon of wave collapse takes place and the light beam blows up in a finite time. This is based on the fact that the Hamiltonian

H = dx |∂ x ψ| 2 -1
2 |ψ| 4 is negative for suitable initial data, then there exists a finite time T such that the quantity dx|∂ x ψ| 2 blows up.

In d spatial dimensions, one consider the generic power nonlinearity equation

i∂ t ψ = -∆ψ -ψ|ψ| p-2 , x ∈ R d .
It is well known that there is locally wellposedness for any data in H s with As long as the well-posedness (local or global) is proved, one want to study the long time behavior of solutions. The initial datum are assumed to be in a Sobolev space H s . These initial datum form different invariant tori. Will these tori survive or be destroyed after a long time ? Can we control the exchange of energy between different modes of these tori ? These questions have been studied extensively recently through the use of KAM theory and Birkhoff normal form theory.

     s ≥ 0 if p ≤ 2 + 4 d s ≥ s * , s * defined by p = 2 + 4 d-2s * if p > 2 + 4 d . Moreover,

The abcd Boussinesq system

In 1757, Euler introduced a set of quasi-linear hyperbolic equations to describe the irrotational waves on the surface of an inviscid fluid under the gravity force. These equations take into account the conservation of mass, momentum and energy while dissipative and surface tension effects are safely ignored. However, in many theoretical, numerical and practical situations, the full Euler equations seem to be more complicated than necessary and further approximated models have been introduced to restricted physical regimes.

In 1872, Joseph Boussinesq, in response to an observation by John Scott Russell of the Introduction solitary wave, derived approximations valid for weakly non-linear, small amplitude and fairly long waves in a channel of approximately constant depth h. The approximations are counterparts to the Stokes expansion, which is appropriate for short waves. The Boussinesq systems take into account the vertical structure of horizontal and vertical flow velocity. Denote A the wave amplitude and the wavelength, the considered situation is that

α := A h 1, β := h 2 2 1, S := α β = A 2 h 3 ≈ 1.
Boussinesq derived the one-dimensional wave model

w tt = w xx + (w 2 ) xx + w xxxx
or its regularized version

w tt = w xx + (w 2 ) xx + w xxtt
and system of two coupled equations

η t + w x + (wη) x = 0 w t + η x + ww x + 1 3 η xtt = 0
or its regularized version

η t + w x + (wη) x = 0 w t + η x + ww x - 1 3 w xxt = 0.
These equations were derived directly from the Eulerian equation of the water wave problem. These equations are formally comparable to KDV equations and Kadomtsev-Petviashvili equations. Indeed, there are an overwhelming number of different but formally Boussinesqtype system. These systems may have different mathematical properties.

In [BCS02 ; BCS04], Bona, Chen and Saut derived a family of Boussinesq-type systems, which depend on four parameters a, b, c, d and are called the abcd-Boussinesq systems

     (1 -b∂ xx )∂ t η + ∂ x (a∂ xx u + u + uη) = 0 (1 -d∂ xx )∂ t u + ∂ x (c∂ xx η + η + 1 2 u 2 ) = 0
Introduction at the point x and time t. The variable u(x, t) is proportional to the horizontal velocity at the height θh, where θ is a fixed constant in the interval [0, 1] and h is the undisturbed water depth. The four parameters a, b, c, d obey the relations

           a + b = 1 2 (θ 2 -1 3 ) c + d = 1 2 (1 -θ 2 ) ≥ 0 a + b + c + d = 1 3 (1.1.4) When (a = b = c = 0, d = 1
3 ), (1.1.3) is the classical Boussinesq system. Higher order nonlinearity system were also derived in

= c = 0, d = 1 3 ), Kaup system (a = 1 3 , b = c = d = 0), coupled BBM system (a = c = 0, b = d = 1 6 ), coupled KdV system (a = c = 1
3 , b = d = 0), etc. All these models are derived from the full Euler equations for two-dimensional water waves under the force of gravity by truncating a Taylor expansion of the velocity potential. As any PDEs for physical regimes, there arise questions, both theoretical and practical : Well-posedness of initial-value problems, existence of solitary-wave solutions, energy exchange in different Fourier modes, etc.

Results of the thesis 1.2.1 An unstable three dimensional KAM torus for the quintic

NLS on the circle

In chapter 2, we prove a KAM result for the quintic NLS in the circle [N19]. The work presented here is the center of an article published in "Dynamics of Partial Differential Equations (DPDE)". We consider the quintic nonlinear Schrödinger equation on the torus

i∂ t u + ∂ xx u = |u 4 |u, (t, x) ∈ R × T.
(1.2.1) associated with the Hamiltonian

h = T |u x | 2 + 1 3 |u| 6 dx
Introduction and the symplectic form -idu ∧ du. The solution u(t) preserves the mass and the momentum

L = T |u| 2 dx, M = T Im(u • ∇u)dx.
Let us expand u and u in Fourier variables :

u(t, x) = j∈Z a j (t)e ijx , u(t, x) = j∈Z b j (t)e -ijx .
The Hamiltonian h of the system reads :

h = j∈Z j 2 a j b j + 1 3 j, ∈Z 3 ;M(j,l)=0 a j 1 a j 2 a j 3 b 1 b 2 b 3 = N + P,
with the symplectic structure -i j∈Z da j ∧ db j , here M(j, l) = j 1 + j 2 + j 3 -1 -2 -3 denotes the momentum of the monomial a j 1 a j 2 a j 3 b 1 b 2 b 3 . We are interested in the dynamic behavior near to 0 of solution of (2.1.1) in two specific forms :

u(t, x) = a p (t)e ipx e -ip 2 t + a q (t)e iqx e -iq 2 t + O(ε), (1.2.2)
and u(t, x) = a p (t)e ipx e -ip 2 t + a q (t)e iqx e -iq 2 t + a m (t)e imx e -im 2 t + O(ε), (1.2.3) or more precisely, the persistence of two and three dimensional linear invariant tori :

T 2 c (p, q) = {|a p | 2 = c 1 , |a q | 2 = c 2 }, (1.2.4) T 3 c (p, q, m) = {|a p | 2 = c 1 , |a q | 2 = c 2 , |a m | 2 = c 3 }, (1.2.5) with 0 < c 1 , c 2 , c 3 1.
We begin by demonstrating that all two dimensional tori are linearly stable.

Theorem 1.2.1. Fix p, q ∈ Z, and s > 1 2 . There exists ν 0 > 0, and for 0 < ν < ν 0 , there exists

D ν ⊂ [1, 2] 2 asymptotically of full measure (i.e., meas([1, 2] 2 \ D ν ) → 0 when ν → 0) such that for ρ ∈ D ν , equation (1.2.1) admits a solution of the form u(x) = j∈Z a j (tω)e ijx where {a j } j is analytic function from T 2 to 2 s satisfying uniformly in θ ∈ T 2 ||a p | - √ νρ 1 | 2 + ||a q | - √ νρ 2 | 2 + j =p,q (1 + j 2 ) s |a j | 2 = O(ν 2 ).

Introduction

Here ω is a nonresonant vector in R 2 that satisfies

ω = (p 2 , q 2 ) + O(ν 2 ).
Furthermore, this solution is linearly stable.

Here we say a solution u is linearly stable if the linearization of the equation at this solution has linear operator whose spectrum contains only pure imaginary eigenvalues.

In the case of three dimensional tori, we will give an example of (p, q, m) and ρ such that for ν small enough the torus

T 3 νρ (p, q, m) = {|a p | 2 = νρ 1 , |a q | 2 = νρ 2 , |a m | 2 = νρ 3 } is linearly unstable. Let = 10 -2 , denote D = D 2 = [2 -, 2 + ] × [1 -, 1 + ] × [9 -, 9 + ].
Then we have the following theorem Theorem 1.2.2. Fix p = -3, q = 10, m = -6, and s > 1 2 . There exists ν 0 > 0, and for 0 < ν < ν 0 , there exists D ν ⊂ D asymptotically of full measure (i.e., meas(D \ D ν ) → 0 when ν → 0) such that for ρ ∈ D ν , equation (1.2.1) admits a solution of the form

u(x) = j∈Z a j (tω)e ijx (1.2.6)
where {a j } j is analytic function from T 3 to 2 s satisfying uniformly in θ ∈ T 3

||a p | - √ νρ 1 | 2 + ||a q | - √ νρ 2 | 2 + ||a m | - √ νρ 3 | 2 + j =p,q,m (1 + j 2 ) s |a j | 2 = O(ν 2 ). (1.2.7)
Here ω is a non resonant vector in R 3 that satisfies

ω = (3 2 , 10 2 , 6 2 ) + O(ν 2 ).
Furthermore, this solution is linearly unstable.

Avoiding the case (2.1.7), we can generalize the theorem for all sets of three Fourier modes (p, q, m) which satisfy the system

     2p + q = m + s + t 2p 2 + q 2 = m 2 + s 2 + t 2 .
(1.2.8)

for some integer numbers s, t.

Scheme of the proof

Introduction

Our results base on a Birkhoff normal form procedure and a KAM theorem. In Birkhoff normal form step, one kills the nonresonances of the quintic nonlinearity P. More precise, the original Hamiltonian is transformed into the following form using a canonical transformation

h = h • τ = N + Z 6 + R 10 ,
where -N is the term N (I) = j∈Z j 2 |a j | 2 ; -Z 6 is a homogeneous polynomial of degree 6 containing only the resonant part

Z 6 = R a j 1 a j 2 a j 3 b 1 b 2 b 3 where R = {(j, ) ∈ Z 3 × Z 3 s.t j 1 + j 2 + j 3 = 1 + 2 + 3 , j 2 1 + j 2 2 + j 2 3 = 2 1 + 2 2 + 2 3 }; -R 10
is the remainder of order 10.

In the KAM procedure, let us write the Hamiltonian as following

h = h 0 + f h 0 = Ω(ρ) • r + a∈Z Λ a (ρ)|ζ a | 2 .
Here ρ is a parameter in D, which is compact in the space R n ; r ∈ R n and θ ∈ T n are the action and angle associated with the internal modes (r, θ) ∈ (R n × T n , dr ∧ dθ) ; ζ = (ζ a ) a∈Z ∈ C Z are the external modes endowed with the standard complex symplectic structure -idζ ∧ dη, with η = ζ. Denote also ω = (ζ, η) -the mappings

Ω : D → R n ,
(1.2.9)

Λ a : D → C, a ∈ Z, (1.2.10)
are smooth ; f = f (r, θ, ζ; ρ) is a perturbation, small compare to the integrable part h 0 .

For the perturbation f, we define its jet function f T (x) as the following Taylor polynomial Introduction of f at r = 0 and ω = 0

f T (x) = f (0, θ, 0) + d r f (0, θ, 0) • r + d ω f (0, θ, 0)[ω] + 1/2d 2 ω f (0, θ, 0)[ω, ω].
By applying iteratively a KAM scheme, we put the Hamiltonian into a normal form

(h 0 + f ) • Φ = h + g with h = Ω(ρ) • r + ζ L , Q(ρ)η L + 1/2 ω F , K(ρ)ω F on normal form
, and the jet part of g is vanishing, i.e., g T ≡ 0. The set L corresponds to elliptic directions, while the set F corresponds to hyperbolic directions. The considered torus is linearly stable if and only if F = ∅. The KAM procedure requires hypotheses on small divisors, the conservation of the mass and the momentum give us a good tool to estimate them. This is verified precisely in the appendix.

Reducibility of Schrödinger equation on a Zoll manifold with unbounded potential

In chapter 3, we are interested in the reducibility of Schrödinger equation on a Zoll manifold [START_REF] Feola | Reducibility of Schrödinger equation on a Zoll manifold with unbounded potential[END_REF]. Precisely, we prove a reducibility result for the linear Schrödinger equation on a Zoll manifold with quasi-periodic in time pseudo-differential perturbation of order less or equal than 1 2 . The work presented here has been published in the "Journal of Mathematical Physics".

In fact, we are considering the following linear Schrödinger equation

i∂ t u = ∆ g u + εW (ωt)u, (t, x) ∈ R × M n (1.2.11)
where ε > 0 is a small parameter, ω ∈ R d , d ≥ 1, is a frequency vector, M n is a Zoll manifold and ∆ g is the positive Laplace-Beltrami operator defined on M n . We recall that a Zoll manifold of dimension n ∈ N is a compact Riemannian manifold such that all the geodesic curves have the same period T, assuming T = 2π. A typical example of Zoll manifold is the sphere S n . The linear operator W is a pseudo-differential operator of order δ ≤ 1 2 . We denote A m the class of pseudo-differential operators of order m ∈ R, then W ∈ C ∞ (T d , A δ ). We consider the solutions in the Sobolev space defined as

H s (M n ) := dom( 1 + ∆ g ) s .
The purpose of this chapter is to find a transformation that puts the non-autonomous equation (1.2.11) into an autonomous form .

Introduction

Our main result is the following.

Theorem 1.2.3. Let 0 < α < 1 and δ ∈ R, δ ≤ 1/2. Assume that the map ϕ → W (ϕ, •) ∈ A δ is C ∞ in ϕ ∈ T d . Then for any s ∈ R, s > n/2 there exists ε 0 > 0 and C > 0 such that, for any 0 < ε ≤ ε 0 there is a set O ε ⊂ [1/2, 3/2] d ⊂ R d with meas([1/2, 3/2] d \ O ε ) ≤ Cε α
(1.2.12) such that the following holds. For any ω ∈ O ε there exists a family of linear isomorphism Ψ(ϕ) ∈ L(H s (M n )) and a Hermitian operator Z ∈ A δ commuting with the Laplacian1 and satisfying

Z L(H s (M n ),H s-δ (M n )) ≤ Cε . (1.2.13) Furthermore • Ψ(ϕ) is unitary on L 2 (M n ) ; • for any n 2 < s ≤ s and any ω ∈ O ε Ψ(ϕ) -Id L(H s (M n ),H s -δ (M n )) + Ψ(ϕ) -1 -Id L(H s (M n ),H s -δ (M n )) ≤ Cε 1-α , Ψ(ϕ) L(H s (M n )) + Ψ(ϕ) -1 L(H s (M n )) ≤ 1 + Cε 1-α , ( 1
i∂ t v = ∆ g v + εZ(v) .
(1.2.15)

As a consequence of reducibility, one proves the existence of almost-periodic solution. Precisely, one has the following corollary.

Corollary 1.2.4. Let W ∈ C ∞ (T d ; A δ ) with δ ≤ 1/2. Then, for any s ∈ R, s > n/2 there exists ε 0 > 0 and C > 0 such that, for any 0 < ε ≤ ε 0 there is a set O ε ⊂ [1/2, 3/2] d ⊂ R d satisfying (1.2.12) such that for any ω ∈ O ε the flow generated by the (1.2.11) equation is bounded in H s (M n ). More precisely, if u 0 ∈ H s (M n ) then there exists a unique solution u ∈ C 1 R ; H s (M n ) of (1.2.11) such that u(0) = u 0 . Moreover, u is almost-periodic in time and satisfies (1 -εC) u 0 H s ≤ u(t) H s ≤ (1 + εC) u 0 H s , ∀ t ∈ R , (1.2.16) Introduction for some C = C(s) > 0.
We also mention the reducibility result on the torus T n in [START_REF] Bambusi | Reducibility of Non-Resonant Transport Equation on with Unbounded Perturbations[END_REF], where the authors study the transport equations on the torus, which is an integrable system, and the recent reducibility result for the Schrödinger equation on the Sphere [START_REF] Feola | Reducibility of Schrödinger equation on the sphere[END_REF], where the authors consider quasi-periodic in time odd perturbations of order < 1/2 and, in particular, do not require pseudo-differential calculus.

Our result gives an idea to approach the nonlinear Schrödinger equation. Consider the non linear Schrödinger equation

i∂ t u = ∆ g u + mu + ε|u| 2 u , u = u(t, x) , t ∈ R , x ∈ M n ,
we would like to solve this equation using Newton's method. Starting with an approximate solution u 0 , we linearize the NLS equation around u 0 and solve the linear equation to obtain u 1 , do this again to obtain u 2 and iterate this procedure to obtain a convergent sequence of solutions. However, this approach has to face some obstacles, which we mention in chapter 3. First, linearizing the cubic nonlinearity at the point u in the direction h, one obtains 2|u| 2 h + u 2 h. As first step of the regularization procedure, one need to eliminate h. The major problems regard the minimal regularity of the potential W (ωt), which is now |u 0 (t)| 2 , and the study of the small divisors relating in KAM procedure.

In the linear Schrödinger equation, we need some requirements for the regularity of the potential and small divisors, which do not persist in Newton scheme.

Scheme of the proof

The result is proven in two steps : regularization step and KAM step. In regularization step, the pseudo-differential calculus is used to transform time-dependent, unbounded potential system in a system with a time-dependent, smoothing perturbation. Then we use a KAM procedure on infinite dimensional matrices to put the equation into an autonomous form.

In the regularization step, we prove that we can transform (by using a symplectic map : u = Φ(v)) the original Schrödinger equation into a new one

i∂ t v = ∆ g v + ε(Z + R(ωt))v , (1.2.17)
where Z is a pseudo-differential operator of order δ independent on time and commuting with ∆ g and R is a ρ-regularizing operator in L(H s (M n ), H s+ρ (M n )) with ρ arbitrary large.

In fact, the regularization step consists of two parts : averaging the pseudo-differential operators, the averaged operators correspond to diagonalized block matrices, and eliminating The idea of averaging pseudo-differential operators is based on the fact that we can write ∆ g = K 0 -Q where Q is a pseudo-differential operator of order -1 chosen (following [Colin]) in such a way that the spectrum of K 0 is included in N + λ for some constant λ ∈ R + (λ > 0). This property makes the K 0 flow periodic and leads us to the fact that if A is a pseudodifferential operator, then its average with respect to the flow of K 0 is given by A = 1 2π 2π 0 e -iτ K 0 Ae iτ K 0 dτ . This idea was already used in a pioneering work of Weinstein [START_REF] Weinstein | Asymptotics of eigenvalue clusters for Laplacian plus a potential[END_REF]. Let us see how this works for the original Schrödinger equation. Let us write

H = H 0 + V (t) where H 0 = ∆ g , and V (t) = εW (ωt) is a pseudo-differential operator of order δ. Denote Y = 1 2π 2π 0 τ (V -V )(τ )dτ ∈ A δ , and S = 1 4 (Y K -1 0 + K -1 0 Y
) a pseudodifferential operator of order δ -1, then S solves the following homological equation

V + i[S, K 2 0 ] = V + order δ -2.
Then the flow generated by S Ψ S = e iS(t) conjugates the original Hamiltonian H to a new one H + (t) with

H + = H 0 + V (t) + order δ -ν
where ν = min(1, 2 -δ). Thus if δ < 2, we have a better equation. In the time eliminating step, we find a Lie transformation Ψ T = e iT that kills the time in Z = V . This time eliminating step requires a non resonance hypothesis on the frequency vector ω of form

| ω • k |≥ γ |k| α k ∈ Z n .
We then alternate the averaging procedure with the time elimination procedure.

After the regularizing procedure, we do a KAM procedure to kill the remainder term R in (1.2.17) which still depends on time but is now a arbitrary smoothing operator. We coincide the operator Z, and the remainder term R with their matrix representation. The matrix Z is block-diagonal, and thus, the spectrum of ∆ g + Z preserves the cluster structure inherited from ∆ g on the Zoll manifold. We also have a link between ρ-smoothing operators, and β-regularizing matrices. The KAM procedure consists in solving homological equations, which are solved blockwise. The increasing size of the blocks may generate loss of regularity, but this loss is acceptable since R is a regularizing operator. We also notice that the new remainder term R + after a KAM step is estimated by a tame inequality with two different norms, a low s-decay norm and a high s + b-decay norm. This Introduction tame estimation allows to obtain a convergent scheme for the sequence of remainders R k .

Birkhoff normal form for abcd Boussinesq system on the circle

In chapter 4, we investigate the long time behavior of abcd Boussinesq system on the circle [N21]. Precisely, we consider the system

     (1 -b∂ xx )∂ t η + ∂ x (a∂ xx u + u + uη) = 0 (1 -d∂ xx )∂ t u + ∂ x (c∂ xx η + η + 1 2 u 2 ) = 0 (t, x) ∈ R × T. (1.2.18)
Where η, u are real functions with zero average

T η(t, x) dx = T u(t, x) dx = 0.
The system was derived by Bona, Chen and Saut [BCS02 ; BCS04], in the vein of the original Boussinesq system, to describe the two dimensional, incompressible and irrotational water wave in the shallow water regime. The two functions η(x, t) and u(x, t) describe the behavior of water in the vertical direction and horizontal direction at the position x and at time t. Four parameters a, b, c, d satisfy the consistency conditions (1.1.4). In fact, the system (1.2.18) has different properties when the four parameters a, b, c, d vary.

In this chapter, we study the system in the "generic Hamiltonian" case, namely the case

b = d > 0, a, c < 0.
Expand the solution in Fourier variables, one has

u(x) = k∈Z * u k e 2iπkx , η(x) = k∈Z * η k e 2iπkx (1.2.19) note that u k = u -k , η k = η -k since u and η are real, then (1.2.18) reads 2      ∂ t η k = -i2πk 1+4π 2 bk 2 ((1 -4π 2 ak 2 )u k + j+l=k u j η l ) ∂ t u k = -i2πk 1+4π 2 bk 2 ((1 -4π 2 ck 2 )η k + 1 2 j+l=k u j u l ) k ∈ Z * . (1.2.20)
The couple solutions (η, u) can be identified with their Fourier expansions (η, u) = (η k , u k ) k∈Z * . We study these solutions on the Sobolev space (s ≥ 0)

Introduction H s := {z = (u, v) = (u k , v k ) k∈Z * ∈ 2 (T) × 2 (T) | z 2 s = k∈Z * |k| 2s (|u k | 2 + |v k | 2 ) < ∞}.
(1.2.21)

One prove that for a long time, the energy exchanges mainly in same Fourier modes η k of horizontal velocity and Fourier modes u k of the depth. Precisely, one has the following informal theorem Theorem 1.2.5. Let fix r ≥ 1, fix b = d > 0, for any sufficiently large s and for almost all values of a, c, assume that the initial data (u, v)(x, 0) s = µ is small, i.e., µ 1,

then there exists a constant C = C(r, s, b) such that ∂ t (α 2 k |u k | 2 + α -2 k |η k | 2 ) ≤ µ r+ 5 2 for t ≤ Cµ -r+ 3 2 Here α k = ( 1-4π 2 ak 2 1-4π 2 ck 2 ) 1 4 . And thus, k∈Z * (α 2 k |u k | 2 + α -2 k |η k | 2 ) ≤ 2µ for t ≤ Cµ -r+ 3 2 .
This result is in fact a corollary of a Birkhoff normal form result stated in chapter 4. The appearance of the scalar α k is unusual. In fact, before stating the Birkhoff normal form theorem, we need to conjugate the original solutions to a new form, which follows a more convenient Hamiltonian equation. One has

ψ k = 1 √ 2 (α k u k + α -1 k η k ), φ k = 1 √ 2 (α k u k -α -1 k η k )
Then the system can be written as

       ∂ t ψ k = -iD k ∇ ψ -k H ∂ t φ k = iD k ∇ φ -k H k ∈ Z * , (1.2.22)
where H is the Hamiltonian

H = H 0 + P with 3 H 0 = k∈N * ω k (|ψ k | 2 + |φ k | 2 ) P = 1 4 √ 2 j+l+h=0 α h α -1 j α -1 l (ψ j + φ j )(ψ l + φ l )(φ h -ψ h )
where

D k = 2πk 1 + 4bπ 2 k 2 , ω k = (1 -4aπ 2 k 2 )(1 -4cπ 2 k 2 ).
Each couple Fourier modes (ψ k , φ k ) is associated with a frequency

Ω k = D k ω k = 2πk 1 + 4bπ 2 k 2 pk 4 + ek 2 + 1 where p = 16π 4 ac, e = -4π 2 (a + c). Since b = d is fixed and a + b + c + d = 1 3 , one has that e = 4π 2 (2b -1 3 ) is fixed and p is bounded in a segment I b := (0, 16π 4 (b -1 6 ) 2 ). Denote B s (µ) a ball of radius µ in H s norm, center at origin and N k := |ψ k | 2 + |φ k | 2 the
H • T = H 0 + Z + R. (1.2.23)
here Z is a polynomial of degree at most r + 2 that commutes with the actions N k , i.e.,

{Z, N k } = 0, ∀k ∈ Z * (1.2.24) and R ∈ C ∞ (B s (µ)) fulfills the estimate sup (ψ,φ) s≤µ/3 X R s ≤ Cµ r+ 3 2 .
(1.2.25)

3. N * = N \ {0}

Introduction

The canonical transformation and its inverse are close to identity

sup (φ,ψ) s≤µ/3 (φ, ψ) -T (φ, ψ) s ≤ C s µ 2 (1.2.26) sup (φ,ψ) s≤µ/3 (φ, ψ) -T -1 (φ, ψ) s ≤ C s µ 2 .
(1.2.27)

Scheme of the proof

The idea to prove Birkhoff normal form result is to search for iterative changes of variables T n = Φ t χn | t=1 , 1 ≤ n ≤ r that put the original Hamiltonian into better and better forms. That is

H • T n = H 0 + Z n + R n+1 .
(1.2.28)

Where Z n is a normal form, which does not affect to the behavior of the solutions, and R n+1 is a perturbation term which is of the size µ n+ 3 2 . At each step of Birkhoff normal form procedure, one need to solve a homological equation

{H 0 , χ} + Z = f (1.2.29)
with Z is in a normal form and f is a polynomial remainder term. We expand f in Taylor series

f (φ, ψ) = j,l f j,l Π k,h ψ j k k φ l h h
and similarly for χ, Z. The homological equation becomes

(Ω(j -l))χ j,l + Z j,l = k∈N * (Ω k (j k -j -k -l k + l -k ))χ j,l + Z j,l = f j,l .
(1.2.30)

Here we use

{H 0 , ψ j k k φ l h h } = (Ω k j k -Ω h l h )ψ j k k φ l h h , Ω k = -Ω -k .
The result is based on a so-called tame inequality

uv s ≤ C( u s v 1 + u 1 v s )
for some constant C > 0 and s. This inequality was introduced in [START_REF] Bambusi | Birkhoff normal form for partial differential equations with tame modulus[END_REF],see also [START_REF] Gérard | Opérateurs pseudo-différentiels et théorème de Nash-Moser[END_REF][START_REF] Hormander | The boundary problems of physical geodesy[END_REF], where the authors proved Birkhoff normal form for many partial differential equations. This inequality allows us to ignore all monomials with more than two high modes. Indeed, consider a function u depending on only high modes, i.e., u = k≥N u k e ikx with Introduction a large number N , one has

u 1 ≤ u s N s-1 .
In our procedure, the polynomial f satisfies the tame inequality. Indeed, according to [START_REF] Bambusi | Birkhoff normal form for partial differential equations with tame modulus[END_REF], it is enough to prove the tame property of the original nonlinearity f = P.

For monomials containing at most two high modes, one proves a nonresonant condition for frequencies, that is

|Ω(j -l)| ≥ κ N α
for some constants κ, α and N. Consider r frequencies Ω j 1 , . . . , Ω jr with j 1 < j 2 < . . . < j r ≤ N, as functions of p, then the corresponding determinant is bounded from below by 1 N α with a constant α = α(r). Combine this with a theorem introduced in [XYQ97], which says that if |g (r) (p)| ≥ d then |g(p)| ≥ h except for a small set of p, one has an estimate for small divisor Ω(j -l) for most value of p.

This nonresonant condition allows us to estimate the solution χ and Z of the homological equation and continue the Birkhoff procedure. Precisely, one has

|X χ | s,R ≤ C N α κ |X f | s,R , |X Z | s,R ≤ C |X f | s,R . (1.2.31)
for some constant C. Here X χ denotes the vector field generated by χ and | • | s,R denotes a norm in Sobolev space of the vector field with variables bounded in the ball with center in the origin and radius R.

2

An unstable three dimensional KAM torus for the quintic NLS

Introduction

We consider the nonlinear Schrödinger equation on the torus

i∂ t u + ∂ xx u = |u| 4 u, (t, x) ∈ R × T (2.1.1)
where T = R/Z. This is an infinite dimensional dynamic system on the phase space (u, u) ∈ L 2 (T) endowed with the symplectic form -idu ∧ du. The flow u(t) preserves the Hamiltonian

h = T |u x | 2 + 1 3 |u| 6 dx,
and also, the mass and the momentum

L = T |u| 2 dx, M = T Im(u • ∇u)dx.
Let us expand u and u in Fourier basis :

u(t, x) = j∈Z a j (t)e ijx , u(t, x) = j∈Z b j (t)e -ijx .
In this variables, the symplectic structure becomes -i j∈Z da j ∧ db j . The Hamiltonian h of the system reads

h = j∈Z j 2 a j b j + 1 3 j, ∈Z 3 ;M(j,l)=0 a j 1 a j 2 a j 3 b 1 b 2 b 3 = N + P,
and the mass and the momentum

L = j∈Z a j b j , M = j∈Z ja j b j ,
Chapitre 2 -An unstable three dimensional KAM torus for the quintic NLS here M(j, l) = j 1 + j 2 + j 3 -1 -2 -3 denotes the momentum of the monomial

a j 1 a j 2 a j 3 b 1 b 2 b 3 .
We can rewrite equation (2.1.1) into a system of infinite number of equations

     i ȧj = j 2 a j + ∂P ∂b j j ∈ Z, -i ḃj = j 2 b j + ∂P ∂a j j ∈ Z.
(2.1.2)

In this article, we are interested in the dynamic behavior near to 0 of solution of (2.1.1) in two specific forms :

u(t, x) = a p (t)e ipx e -ip 2 t + a q (t)e iqx e -iq 2 t + O(ε), (2.1.3) and u(t, x) = a p (t)e ipx e -ip 2 t + a q (t)e iqx e -iq 2 t + a m (t)e imx e -im 2 t + O(ε), (2.1.4)
or more precisely the persistence of two and three dimensional linear invariant tori :

T 2 c (p, q) = {|a p | 2 = c 1 , |a q | 2 = c 2 }, (2.1.5) T 3 c (p, q, m) = {|a p | 2 = c 1 , |a q | 2 = c 2 , |a m | 2 = c 3 }, (2.1.6) with 0 < c 1 , c 2 , c 3 1.
The first result of this paper is stated for two dimensional tori.

Theorem 2.1.1. Fix p, q ∈ Z, and s > 1 2 . There exists ν 0 > 0, and for 0 < ν < ν 0 , there exists

D ν ⊂ [1, 2] 2 asymptotically of full measure (i.e. meas([1, 2] 2 \ D ν ) → 0 when ν → 0) such that for ρ ∈ D ν , equation (2.1.1) admits a solution of the form u(x) = j∈Z a j (tω)e ijx
where {a j } j are analytic functions from T 2 to 2 s satisfying uniformly in θ ∈ T 2

||a p | - √ νρ 1 | 2 + ||a q | - √ νρ 2 | 2 + j =p,q (1 + j 2 ) s |a j | 2 = O(ν 2 ).
Here ω is a nonresonant vector in R 2 that satisfies

ω = (p 2 , q 2 ) + O(ν 2 ).
Furthermore, this solution is linearly stable.

Remark 2.1.2.

-Here, the notation nonresonant means that there is no {0, 0} = ∈ Zš such that ω • = 0. -u is linearly stable if the linear equation system obtained by linearizing the system (2.1.2) on this solution has the form

  ȧ ḃ  = A   a b  
, where A is a linear operator whose all the eigenvalues have negative real part. By contrast, it is linearly unstable if the spectrum of A contains eigenvalues with positive real part.

For three dimensional tori, it is too complicated 1 to consider the general case. In order to apply KAM theorem 2.2.2, we avoid the case where there is

∈ Z solving equation 2      2j 1 + j 2 = 2j 3 + 2j 2 1 + j 2 2 = 2j 2 3 + 2 .
(2.1.7)

In this paper, we will give here an example of (p, q, m) and ρ such that for ν small enough the torus

T 3 νρ (p, q, m) = {|a p | 2 = νρ 1 , |a q | 2 = νρ 2 , |a m | 2 = νρ 3 } is linearly unstable. For = 10 -2 , denote D = D 2 = [2 -, 2 + ] × [1 -, 1 + ] × [9 -, 9 + ].
Theorem 2.1.3. Fix p = -3, q = 10, m = -6, and s > 1 2 . There exists ν 0 > 0, and for 0 < ν < ν 0 , there exists D ν ⊂ D asymptotically of full measure (i.e. meas(D \ D ν ) → 0 when ν → 0) such that for ρ ∈ D ν , equation (2.1.1) admits a solution of the form

u(x) = j∈Z a j (tω)e ijx (2.1.8)
where {a j } j are analytic functions from T 3 to 2 s satisfying uniformly in θ ∈ T 3

||a p | - √ νρ 1 | 2 + ||a q | - √ νρ 2 | 2 + ||a m | - √ νρ 3 | 2 + j =p,q,m (1 + j 2 ) s |a j | 2 = O(ν 2 ). (2.1.9)
Here ω is a non resonant vector in R 3 that satisfies

ω = (3 2 , 10 2 , 6 2 ) + O(ν 2 ).
Furthermore, this solution is linearly unstable.

In order to prove Theorems 2.1.1, 2.1.3, we follow a general strategy developed in [START_REF] Grébert | Stable and unstable time quasi periodic solutions for a system of coupled NLS equations[END_REF] for a system of coupled nonlinear Schrödinger equations on the torus. Firstly, 1. the difficulty is to verify KAM hypotheses 2. in this case, the linear part a 2 j1 a j2 b 2 j3 b + b 2 j1 b j2 a 2 j3 a of the mode would create the instability, and the energy would soon transfer mainly between four modes p, q, m, , which was studied carefully in [START_REF] Grébert | Resonant dynamics for the quintic nonlinear Schrödinger equation[END_REF].

we apply a Birkhoff normal form procedure (Proposition 2.3.1) to kill the nonresonance of P . Then we use sympletic changes of variables to diagonalize the effective part into the form of h 0 . The main difference between Theorem 2.1.1 and Theorem 2.1.3 is for the linear stability of the solution, which is explained by the presence of hyperbolic directions on the torus T 3 νρ (-3, 10, -6). In section 3, we will see that in this case the energy will drain out of these three modes into two exterior modes {9, 1}. Since the proof bases on KAM theorem in [START_REF] Grébert | Stable and unstable time quasi periodic solutions for a system of coupled NLS equations[END_REF], readers are suggested to take a look at the original statement for further understanding.

The study of finite dimensional tori in an infinite dimensional phase space was pioneered by J. Bourgain [START_REF] Bourgain | Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations[END_REF] in 1988. However, the existence of unstable KAM tori in one dimensional context was first proved by B. Grébert and V. Vilaça da Rocha [START_REF] Grébert | Stable and unstable time quasi periodic solutions for a system of coupled NLS equations[END_REF] in 2017, where they studied the system of coupled nonlinear Schrödinger equations on the torus. For the equation (2.1.1), in case of u(0, x) supported mainly in four modes (p, q, m, s), which satisfy such a relation in (2.1.7), the study of solution was studied carefully in [START_REF] Grébert | Resonant dynamics for the quintic nonlinear Schrödinger equation[END_REF] and [START_REF] Haus | KAM for beating solutions of the quintic NLS[END_REF]. In particular, in [START_REF] Haus | KAM for beating solutions of the quintic NLS[END_REF] they proved the recurrent exchange of energy between those modes.

Acknowledgement : I would like to thank Professor Benoît Grébert for motivating me to publish this paper with numerous suggestions and discussions. I also would like to thank my friends Le Quoc Tuan and Nguyen Thi Lan Anh for computations in the appendix A.

KAM theorem

In order to proof Theorems 2.1.1 and 2.1.3, we recall a KAM theorem stated in [START_REF] Grébert | Stable and unstable time quasi periodic solutions for a system of coupled NLS equations[END_REF]. We consider a Hamiltonian h = h 0 + f, where h 0 is a quadratic Hamiltonian in normal form

h 0 = Ω(ρ) • r + a∈Z Λ a (ρ)|ζ a | 2 .
(2.2.1)

Here -ρ is a parameter in D, which is a compact in the space R n ; -r ∈ R n
d j ≤ d < ∞. If a ∈ L j , we denote [a] = L j and w a = j, for a ∈ F we set w a = 1; -the mappings Ω : D → R n , (2.2.2) Λ a : D → C, a ∈ Z, (2.2.3) are smooth ; -f = f (r, θ, ζ; ρ) is a perturbation, small compared to the integrable part h 0 .
Linear space Let s ≥ 0, we consider the complex weighted 2 -space

Z s = {ζ = (ζ a ∈ C, a ∈ Z) | ζ s < ∞},
where

ζ s = a∈Z |ζ a | 2 w 2s a .
Similarly we define

Y s = {ζ L = (ζ a ∈ C, a ∈ L) | ζ L s < ∞},
with the same norm. We endow Z s ×Z s and Y s ×Y s with the symplectic structure -idζ ∧dη, with η = ζ.

A class of Hamiltonian functions. Denote ω = (ζ, η). On the space

C n × C n × (Z s × Z s ) we define the norm (r, θ, ω) s = max (|r|, |θ|, ζ s ) .
For σ > 0 we denote

T n σ = {θ ∈ C n : |Im θ| < σ}/2πZ n .
For σ, µ ∈ (0, 1] and s ≥ 0 we set

O s (σ, µ) = {r ∈ C n : |r| < µ 2 } × T n s × {ω ∈ Z s × Z s : ζ s < µ}.
Chapitre 2 -An unstable three dimensional KAM torus for the quintic NLS

We will denote points in O s (σ, µ) as x = (r, θ, ω). Let f : O 0 (σ, µ) × D → C be a C 1 ˘function 3 , real holomorphic in the first variable x, such that for all ρ ∈ D, x ∈ O s (σ, µ) :

∇ ω f (x, ρ) ∈ Z s × Z s and ∇ 2 ω L ω L f (x, ρ) ∈ L(Y s , Y s )
are real holomorphic functions. We denote by T s (σ, µ, D) this set of functions. For f ∈ T s (σ, µ, D), we define

|∂ j ρ f | σ,µ,D = sup x∈O s (σ,µ); ρ∈D max(|∂ j ρ f |, µ ∂ j ρ ∇ ω f (x, ρ) s , µ 2 ∇ 2 ω L ω L ∂ j ρ f (x, ρ) ),
and

[f ] s σ,µ,D = max j=0,1 (|∂ j ρ f | σ,µ,D ).
Jet functions For any f ∈ T s (σ, µ, D), we define its jet f T (x) as the following Taylor polynomial of f at r = 0 and ω = 0

f T (x) = f (0, θ, 0) + d r f (0, θ, 0) • r + d ω f (0, θ, 0)[ω] + 1/2d 2 ω f (0, θ, 0)[ω, ω].
Infinite matrices For the elliptic variables, we denote by M s the set of infinite matrices A : L × L → C such that A maps linearly Y s into Y s . We provide M s with the operator norm

|A| s = A L(Ys,Ys) .
We say that a matrix A ∈ M s is on normal form if it is block diagonal and Hermitian, i.e.

A β α = 0 for [α] = [β] and A β α = A α β for α, β ∈ L.
In particular, if A ∈ M s is on normal form, its eigenvalues are real.

Normal form A quadratic Hamiltonian function is on normal form if it reads

h = Ω(ρ) • r + ζ L , Q(ρ)η L + 1/2 ω F , K(ρ)ω F
3. C 1 regularity with respect to ρ in the Whitney sense for some vector function Ω(ρ) ∈ R n , some matrix functions Q(ρ) ∈ M s on normal form and K(ρ) is a matrix F × F → C symmetric in the following sense :K β α = t K α β . Poisson brackets The Poisson brackets of two Hamiltonian functions is defined by

{f, g} = ∇ θ f • ∇ r g -∇ r f • ∇ θ g -i ∇ ω f, J∇ ω g .
Remark 2.2.1. A function f is preserved under the flow u(t) if and only if it commutes with h i.e. {f, h} = 0. By this, we have

{L, h} = {M, h} = 0.
Hypothesis A0 There exists a constant C > 0 such that

|Λ a -|w a | 2 | ≤ C, ∀a ∈ L.
Hypothesis A1 There exists δ > 0 such that

|Λ a | ≥ δ, ∀a ∈ L; |Im Λ a | ≥ δ, ∀a ∈ F; |Λ a -Λ b | ≥ δ, ∀a, b ∈ Z, [a] = [b]; |Λ a + Λ b | ≥ δ, ∀a, b ∈ L.
Hypothesis A2 There exists δ > 0 such that for all Ω δ˘close to Ω 0 in C 1 norm and for all k ∈ Z n \{0} :

1. either |Ω(ρ) • k| ≥ δ ∀ρ ∈ D, or there exists a unit vector z = z(k) ∈ R n such that (∇ ρ • z) (Ω(ρ) • k) ≥ δ ∀ρ ∈ D; 2. for all a ∈ L either |Ω(ρ) • k + Λ a | ≥ δ ∀ρ ∈ D,
or there exists a unit vector

z = z(k) ∈ R n such that (∇ ρ • z) (Ω(ρ) • k + Λ a ) ≥ δ ∀ρ ∈ D; 3. for all α, β ∈ L and a ∈ [α], b ∈ [β] either |Ω(ρ) • k + Λ a ± Λ b | ≥ δ ∀ρ ∈ D,
or there exists a unit vector

z = z(k) ∈ R n such that (∇ ρ • z) (Ω(ρ) • k + Λ a ± Λ b ) ≥ δ ∀ρ ∈ D; 4. for all a, b ∈ F |Ω(ρ) • k + Λ a ± Λ b | ≥ δ.
Theorem 2.2.2 (KAM theorem). Assume that hypothesis A0, A1, A2 are satisfied, f ∈ T s (σ, µ, D), f commutes with L, M and s > 1/2. Let γ > 0, there exists a constant

C 0 such that if [f ] s σ,µ,D ≤ C 0 δ, ε := [f T ] s σ,µ,D ≤ C 0 δ 1+γ , (2.2.4)
then there exists a Cantor set D ⊂ D asymptotically of full measure (i.e. meas(D\D ) → 0 when ε → 0) and there exists a symplectic change of variables Φ :

O s (σ/2, µ/2) → O s (σ, µ) such that for all ρ ∈ D (h 0 + f ) • Φ = h + g with h = Ω(ρ)•r+ ζ L , Q(ρ)η L +1/2 ω F , K(ρ)ω F on
normal form, and g ∈ T s (σ/2, µ/2, D ) with g T ≡ 0. Furthermore there exists C > 0 such that for all ρ ∈ D

|Ω -Ω 0 | ≤ Cε, |Q -diag (Λ a , a ∈ L) | ≤ Cε, |JK -diag (Λ a , a ∈ F) | ≤ Cε.
As a dynamic consequence Φ ({0} × T n × {0}) is an invariant torus for h 0 + f and this torus is linearly stable if and only if F = ∅ (see [START_REF] Grébert | Stable and unstable time quasi periodic solutions for a system of coupled NLS equations[END_REF] ).

Here, the matrix J is of the form,

  0 -I I 0  
where I is identity matrix of size #F .

Remark 2.2.3. In [START_REF] Grébert | Stable and unstable time quasi periodic solutions for a system of coupled NLS equations[END_REF], they constrained f in a restricted class instead of using commutation of f with L, M since they considered a system of coupled NLS equation with more complicated nonlinearities.

Applications

Applications

The Birkhoff normal form procedure. We recall a result proved in [START_REF] Grébert | Resonant dynamics for the quintic nonlinear Schrödinger equation[END_REF].

Proposition 2.3.1. There exist a canonical change of variable

τ from O s (σ, µ) into O s (2σ, 2µ) such that h = h • τ = N + Z 6 + R 10 ,
where -N is the term N (I) = j∈Z j 2 |a j | 2 ; -Z 6 is the homogeneous polynomial of degree 6

Z 6 = R a j 1 a j 2 a j 3 b 1 b 2 b 3 where R = {(j, ) ∈ Z 3 × Z 3 s.t j 1 + j 2 + j 3 = 1 + 2 + 3 , j 2 1 + j 2 2 + j 2 3 = 2 1 + 2 2 + 2 3 }; -R 10
is the remainder of order 10, i.e a Hamiltonian satisfying X R 10 (x) s ≤ C x 9 s for all x ∈ O s (σ, µ); τ is close to the identity : there exists a constant C such that

τ (x) -x ≤ C x 2 for all x ∈ O s (σ, µ).
Henceforth, since we do not care about constant, we shall write a b in order to say a Cb.

Persistence of 2 dimensional tori.

Firstly, we want to study the persistence of the two dimensional invariant torus

T 2 νρ (p, q) for equation (2.1.1) for ν small. Choose            a p = (νρ 1 + r 1 (t)) 1 2 e iθ 1 (t) =: I p e iθ 1 (t) a q = (νρ 2 + r 2 (t))
1 2 e iθ 2 (t) =: I q e iθ 2 (t)

a j = ζ j j = p, q,
where {ρ 1 , ρ 2 } ∈ [1, 2] 2 = D and ν is a small parameter.The canonical symplectic structure Chapitre 2 -An unstable three dimensional KAM torus for the quintic NLS now becomes

-idζ ∧ dη -dI ∧ dθ with I = (I 1 , I 2 ), θ = (θ 1 , θ 2 ), ζ = (ζ j ) j and η = (η j ) j = (ζ j ) j . Let T lin ρ := {(I, θ, ζ)||I -νρ| = 0, |Im θ| < σ, ζ s = 0}
and its neighborhood

T ρ (ν, σ, µ, s) := {(I, θ, ζ)||I -νρ| < νµ 2 , |Im θ| < σ, ζ s < ν 1/2 µ}.
We want to study the persistence of torus T ρ (ν, σ, µ, s). Indeed we have

T ρ (ν, σ, µ, s) ≈ O s (σ, ν 1/2 µ) = {(r, θ, ζ)||r| < νµ 2 , |Im θ| < σ, ζ s < ν 1/2 µ}.
By Theorem 2.3.1 we have

h • τ = N + Z 6 + R 10 .
We see that the term N contributes to the effective part and the term R 10 contributes to the remainder term f. So we just need to focus on the term Z 6 . Let us split it :

Z 6 = Z 0,6 + Z 1,6 + Z 2,6 + Z 3,6 .
Here Z 0,6 , Z 1,6 , Z 2,6 are homogeneous polynomial of degree 6 which contain respectively external modes of order 0, 1, 2. Z 3,6 is a homogeneous polynomial of degree 6 contains external modes of at least order 3, this term contributes the remainder term. Thank to Lemma 2.2 in [START_REF] Grébert | Resonant dynamics for the quintic nonlinear Schrödinger equation[END_REF], the term Z 1,6 = 0. We have

Z 0,6 = |a p | 6 + |a q | 6 + 9 |a p | 4 |a q | 2 + |a p | 2 |a q | 4 = (νρ 1 + r 1 ) 3 + (νρ 2 + r 2 ) 3 + 9 (νρ 1 + r 1 ) (νρ 2 + r 2 ) (νρ 1 + r 1 + νρ 2 + r 2 ) = ν 3 (ρ 3 1 + ρ 3 2 + 9ρ 2 1 ρ 2 + 9ρ 2 2 ρ 1 ) + 3ν 2 r 1 (ρ 2 1 + 6ρ 1 ρ 2 + 3ρ 2 2 ) + r 2 (ρ 2 2 + 6ρ 1 ρ 2 + 3ρ 2 1 ) + jet free
where the notation "jet free" means that the remaining Hamiltonian has a vanishing jet. For the term Z 2,6 , there are two cases that can happen.

First case

We assume that there is no solution4 {s, t} = {p, q} for

     2p + s = 2q + t 2p 2 + s 2 = 2q 2 + t 2 .
(2.3.1)

Hence

Z 2,6 = Z 1 2,6 = 9 |a p | 4 + |a q | 4 + 4|a p | 2 |a q | 2 j =p,q |a j | 2 = 9ν 2 ρ 2 1 + ρ 2 2 + 4ρ 1 ρ 2 j =p,q |ζ j | 2 +jet free.
Hence

h • τ = h e + R
where the effective Hamiltonian h e reads

h e = p 2 + 3ν 2 ρ 2 1 + 3ρ 2 2 + 6ρ 1 ρ 2 r 1 + q 2 + 3ν 2 ρ 2 2 + 3ρ 2 1 + 6ρ 1 ρ 2 r 2 + j j 2 + 9ν 2 ρ 2 1 + ρ 2 2 + 4ρ 1 ρ 2 |ζ j | 2 = Ω(ρ) • r + j =p,q Λ j |ζ j | 2
where

Ω(ρ) =   p 2 + 3ν 2 (ρ 2 1 + 3ρ 2 2 + 6ρ 1 ρ 2 ) q 2 + 3ν 2 (ρ 2 2 + 3ρ 2 1 + 6ρ 1 ρ 2 )   and Λ j = j 2 + 9ν 2 ρ 2 1 + ρ 2 2 + 4ρ 1 ρ 2 .
The remainder term R reads

R = R 10 + Z 3,6 + 3νρ 1 r 2 1 + r 3 1 + 3νρ 2 r 2 2 + r 3 2 + 9r 1 r 2 (r 1 + r 2 ) + r 2 1 + r 2 2 + 2ν(ρ 1 + 2ρ 2 )r 1 + 2ν(ρ 2 + 2ρ 1 )r 2 j =p,q |ζ j | 2 .
In order to work on O s (σ, µ) we use the rescaling

Ψ : r → νr, ζ → ν 1/2 ζ. (2.3.2)
Chapitre 2 -An unstable three dimensional KAM torus for the quintic NLS

The symplectic structure now becomes

-νdr ∧ dθ -iνdζ ∧ dη.
By definition, this change of variables maps O s (σ, µ) to a neighborhood of T ρ (ν, σ, µ, s). Since τ is close to identity, the change of variables

Φ ρ = τ •Ψ maps O s (σ, µ) to T ρ (ν, 2σ, 2µ, s).
By this change of variables, we have

h • Φ ρ -C = (h e + R) • Ψ = νh 0 + νf
where C is a constant, h 0 and f are defined by

h 0 = 1 ν h e • Ψ f = 1 ν R • Ψ. By Theorem 2.3.1, R 10 ∈ T s (σ, ν 1/2 µ, D).
We check that the remaining part of f is in

T s (σ, µ, D)
. By construction, f commutes 5 with L and M. To estimate the norm of f, notice that R contains only term of order at least 3 in ν and

R T = R T 10 is of order 9/2 in ν, so that [f ] s σ,µ,D ν 2 and [f T ] s σ,µ,D ν 7/2 .

So we have proved :

Theorem 2.3.2. Assume that for p, q ∈ Z there do not exist s, t solving the equation (2.3.1). Then, the change of variables Φ ρ = τ • Ψ is real holomorphic, symplectic and analytically depending on ρ satisfying -Φ ρ : O s (σ, µ) → T ρ (ν, 2σ, 2ν, s); -Φ ρ puts the Hamiltonian h in normal form in the following sense :

1 ν (h • Φ ρ -C) = h 0 + f
where C is a constant and the effective part h 0 of the Hamiltonian reads

h 0 = Ω(ρ) • r + j =p,q Λ j |ζ j | 2
5. since h commutes with L, M and all the changes of variables are symplectic

with

Ω(ρ) =   p 2 + 3ν 2 (ρ 2 1 + 3ρ 2 2 + 6ρ 1 ρ 2 ) q 2 + 3ν 2 (ρ 2 2 + 3ρ 2 1 + 6ρ 1 ρ 2 )   and Λ j = j 2 + 9ν 2 ρ 2 1 + ρ 2 2 + 4ρ 1 ρ 2 ;
-The remainder term f belongs to T s (σ, µ, D) and satisfies

[f ] s σ,µ,D ν 2 and [f T ] s σ,µ,D ν 7/2 .
Second case Assume that there are6 s, t = p, q solving (2.3.1), hence

Z 2,6 = Z 1 2,6 + 9(a 2 p a s b 2 q b t + b 2 p b s a 2 q a t ) = Z 1 2,6 + Z s,t
For the second term, let us rewrite it

9(νρ 1 + r 1 )(νρ 2 + r 2 ) e 2i(θ 1 -θ 2 ) ζ s η t + e -2i(θ 1 -θ 2 ) η s ζ t
The effective part of this term is just given by

9ν 2 ρ 1 ρ 2 e 2i(θ 1 -θ 2 ) ζ s η t + e -2i(θ 1 -θ 2 ) η s ζ t . Notice that {I s , ζ s η t + η s ζ t } = {I t , ζ s η t + η s ζ t } = 0.
This gives us a clue that the above term does not effect to the stability of the solution.

In order to kill the angles, we introduce the symplectic change of variables Ψ angles :

O s (σ, µ) → O s (σ, µ), (r 1 , r 2 , θ, ζ) → (r 1 , r 2 , θ, ζ ) defined by                            ζ s = e 2i(θ 1 -θ 2 ) ζ s ζ t = ζ t ζ j = ζ j , j = s, t, p, q r 1 = r 1 -2|ζ s | 2 r 2 = r 2 + 2|ζ s | 2 .

By this change of variables

h = h • Ψ angles = C + h e + R.
Here C is a constant given by

C = ν 3 (ρ 3 1 + ρ 3 2 + 9ρ 2 1 ρ 2 + 9ρ 2 2 ρ 1 ) + 9(νp 2 ρ 1 + νq 2 ρ 2 ).
The effective Hamiltonian h e reads

h e = p 2 + 3ν 2 ρ 2 1 + 3ρ 2 2 + 6ρ 1 ρ 2 r 1 + q 2 + 3ν 2 ρ 2 2 + 3ρ 2 1 + 6ρ 1 ρ 2 r 2 + j =p,q,s,t j 2 + 9ν 2 ρ 2 1 + ρ 2 2 + 4ρ 1 ρ 2 |ζ j | 2 + t 2 + 9ν 2 ρ 2 1 + ρ 2 2 + 4ρ 1 ρ 2 |ζ t | 2 + s 2 + 2p 2 -2q 2 + ν 2 21ρ 2 2 -3ρ 2 1 + 36ρ 1 ρ 2 |ζ s | 2 + 9ν 2 ρ 1 ρ 2 (ζ s η t + η s ζ t ).
It is on normal form

Ω(ρ) • r + j =p,q,s,t Λ j |ζ j | 2 + Λ s |ζ s | 2 + Λ t |ζ t | 2 + 9ν 2 ρ 1 ρ 2 (ζ s η t + η s ζ t )
where Ω(ρ) and Λ j are defined as in the first case except

Λ s = t 2 + ν 2 21ρ 2 2 -3ρ 2 1 + 36ρ 1 ρ 2 .
In order to diagonalize h e , we use a symplectic change of variables of the form

     ζ t + = 1 √ 1+α 2 (ζ t + αζ s ) ζ t -= 1 √ 1+α 2 (ζ s -αζ t ) with α = -2ρ 2 1 +2ρ 2 2 + √ 4ρ 4 1 +2ρ 2 1 ρ 2 2 +4ρ 4 2 3ρ 1 ρ 2 .
Then h e can be diagonalized as

Ω(ρ) • r + j =p,q,s,t Λ j |ζ j | 2 + Λ t + |ζ t + | 2 + Λ t -|ζ t -| 2 where      Λ t + = Λ t -9ν 2 ρ 1 ρ 2 α Λ t -= Λ s + 9ν 2 ρ 1 ρ 2 α.
The remainder term R reads

R = R 10 • Ψ angles + Z 3,6 • Ψ angles + 3νρ 1 r 2 1 + r 3 1 + 3νρ 2 r 2 2 + r 3 2 + 9r 1 r 2 (r 1 + r 2 ) + r 2 1 + r 2 2 + 2ν(ρ 1 + 2ρ 2 )r 1 + 2ν(ρ 2 + 2ρ 1 )r 2 j =p,q |ζ j | 2 with r 1 = r 1 + 2|ζ s | 2 , r 2 = r 2 -2|ζ s | 2 .
Using the rescaling Ψ introduced in (2.3.2), we get

(h e + R) • Ψ = νh 0 + νf. Since Ψ angles : O s (σ, µ) → O s (σ, 3µ
) and τ is closed to identity, we have τ • Ψ angles • Ψ :

O s (σ, µ) → T ρ (ν, 2σ, 4µ, s).
The study of f is the same as in the previous case. Then we get :

Theorem 2.3.3. Assume that p, q, s, t satisfy the equation 2.3.1. The change of variables Φ ρ = τ • Ψ angles • Ψ is a real holomorphic transformations, analytically depending on ρ satisfying -Φ ρ : O s (σ, µ) → T ρ (ν, 2σ, 4µ, s); -Φ ρ puts the Hamiltonian h in normal form in the following sense :

1 ν (h • Φ ρ -C) = h 0 + f
where C is a constant and the effective part h 0 of the Hamiltonian reads

h 0 = Ω(ρ) • r + j =p,q,s,t Λ j |ζ j | 2 + Λ t + |ζ t + | 2 + Λ t -|ζ t -| 2 with Ω(ρ) =   p 2 + 3ν 2 (ρ 2 1 + 3ρ 2 2 + 6ρ 1 ρ 2 ) q 2 + 3ν 2 (ρ 2 2 + 3ρ 2 1 + 6ρ 1 ρ 2 )   and Λ j = j 2 + 9ν 2 ρ 2 1 + ρ 2 2 + 4ρ 1 ρ 2 ,
Chapitre 2 -An unstable three dimensional KAM torus for the quintic NLS -The remainder term f belongs to T s (1, 1, D) and satisfies

[f ] s σ,µ,D ν 2 and [f T ] s σ,µ,D ν 7/2 .
Now we can finish the proof of Theorem 2.1.1.

Proof of Theorem 2.1.1. By Theorem 2.3.2 and 2.3.3, there exists a symplectic change of variables Φ ρ , on a asymtotical set D ν ⊂ D = [1, 2] 2 that puts the Hamiltonian h = N +P in normal form h 0 + f. In appendix A we verify that h 0 + f satisfies the hypotheses of KAM theorem 2.2.2 for δ = ν 2 , ε = ν 7/2 = δ 7/4 and Ω 0 = ω = (p 2 , q 2 ) + O(ν 2 ). Since the hyperbolic set F is empty, Φ -1 ρ • T lin ρ is an invariant KAM torus that is linearly stable.

Persistence of 3 dimensional tori. Assume that

                   a p = (νρ 1 + r 1 (t)) 1 2 e iθ 1 (t) =: I p e iθ 1 (t)
a q = (νρ 2 + r 2 (t))

1 2 e iθ 2 (t) =: I q e iθ 2 (t) a m = (νρ 3 + r 3 (t))

1 2 e iθ 3 (t) =:

√ I m e iθ 3 (t) a j = ζ j j ∈ Z \ {p, q, m}
where ρ = (ρ 1 , ρ 2 , ρ 3 ) ∈ D ⊂ R 3 and ν is a small parameter. The canonical symplectic structure now becomes

-idζ ∧ dη -dI ∧ dθ with I = (I p , I q , I m ), θ = (θ 1 , θ 2 , θ 3 ), ζ = (ζ j ) j∈Z\{p,q,m} and η = (η j ) j∈Z\{p,q,m} = (ζ j ) j∈Z\{p,q,m} .
The same as in two-modes case, we have

h := h • τ = N + Z 6 + R 10 .
We see that as in the previous case, the term N contributes to the effective Hamiltonian h 0 and the term R 10 contributes to the remainder term f. So we just need to focus on the term Z 6 . Let us split it :

Z 6 = Z 0,6 + Z 1,6 + Z 2,6 + Z 3,6 .
Here, Z 0,6 is a homogeneous polynomial of degree 6 which just contains inner modes (p, q, m) ; Z 1,6 , Z 2,6 are homogeneous polynomials of degree 6 which contain outer modes of order 1 and 2. Z 3,6 is a homogeneous polynomial of degree 6 contains outer modes of at least order 3, this term contributes the remainder term. We have :

Z 0,6 = |a p | 6 + |a q | 6 + |a m | 6 + 9 j, ∈{p,q,m} |a j | 4 |a | 2 + 36|a p | 2 |a q | 2 |a m | 2
Even if it looks a bit more complicated, we deal with Z 0,6 as in the previous case. We assume that there is no solution to (2.1.7), so that Z 1,6 = 0. For Z 2,6 , we have

Z 2,6 = j 1 ,j 2 , |a j 1 | 2 |a j 2 | 2 |a | 2 + s 1 ,t 1 ∈A a 2 j 3 a s 1 b 2 j 4 b t 1 + b 2 j 3 b s 1 a 2 j 4 a t 1 + s 2 ,t 2 ∈B a 2 j 5 a j 6 b j 7 b s 2 b t 2 + b 2 j 5 b j 6 a j 7 a s 2 a t 2 + s 3 ,t 3 ∈C a 2 j 9 a s 3 b j 8 b j 10 b t 3 + b 2 j 9 b s 3 a j 8 a j 10 a t 3 + s 4 ∈E a 2 j 11 a j 12 b j 13 b 2 s 4 + b 2 j 11 b j 12 a j 13 a 2 s 4
with j i ∈ {p, q, m}, s i , t i / ∈ {p, q, m} and s i = t i . The sets A, B, C, E are given by

A ↔      2j 3 + s 1 = 2j 4 + t 1 2j 2 3 + s 2 1 = 2j 2 4 + t 2 1 B ↔      2j 5 + j 6 = j 7 + s 2 + t 2 2j 2 5 + j 2 6 = j 2 7 + s 2 2 + t 2 2 C ↔      2j 9 + s 3 = j 8 + j 10 + t 3 2j 2 9 + s 2 3 = j 2 8 + j 2 10 + t 2 3 E ↔      2j 11 + j 12 = j 13 + 2s 4 2j 2 11 + j 2 12 = j 2 13 + 2s 2 4 .
Assume that A, B, C, E are disjoint 7 i.e. there is no s or t appearing in two of these sets. We shall deal with each term one by one (in case it's not empty).

The first term just depends on the actions, and we have

|a j 1 | 2 |a j 2 | 2 |a | 2 = ν 2 ρ j 1 ρ j 2 |ζ | 2 + jet free.
The second and the fourth term are similar, since their effective parts are all of the form

9e iα ζ s η t + 9e -iα η s ζ t .
7. this is the case for the example considered in Theorem 2.1.3

The idea to deal with these two terms is the same as that in the two-modes case. Since

{I s + I t , ζ s η t } = {I s + I t , ζ t η s } = 0,
these terms do not affect the stability of the flow. Since A, B, C, E are disjoint, and as in the two-modes case, a change of variables that used to deal with a pair s, t only affects that modes, i.e the changes of variables commute. We denote Φ 1 as the composition of all changes of variables used to deal with the sets A and C.

For the third term, its effective parts are of the form

18ν 2 ρ j 5 √ ρ j 6 ρ j 7 (e iα ζ s ζ t + e -iα η s η t )
where α = θ j 7 -θ j 6 -2θ j 5 . For explicitness, we will consider the case j 5 = p, j 6 = q, j 7 = m, and s, t solve the following equation

     2p + q = m + s + t 2p 2 + q 2 = m 2 + s 2 + t 2 , (2.3.3) then α = θ 3 -θ 2 -2θ 1 .
An example for this could be (p, q, m, s, t) = (-3, 10, -6, 1, 9). In order to kill the angles, we introduce the symplectic change of variables Ψ ang,1 :

O s (σ, µ) → O s (σ, 3µ); (r, θ, ζ) → (r , θ, ζ ) defined by                                  ζ s = ie -iα η s η s = ie iα ζ s ζ t = ζ t η t = η t ζ j = ζ j , η j = η j j = s, t, p, q r 1 = r 1 + 2|ζ s | 2 r 2 = r 2 + |ζ s | 2 , r 3 = r 3 -|ζ s | 2 .
The effective part related to s, t is of the form

Λ s |ζ s | 2 + Λ t |ζ t | 2 -18iν 2 ρ 1 √ ρ 2 ρ 3 (ζ s η t + η s ζ t ) (2.3.4)
where

Λ t = t 2 + 9ν 2 (ρ 2 1 + ρ 2 2 + ρ 2 3 + 4ρ 1 ρ 2 + 4ρ 2 ρ 3 + 4ρ 3 ρ 1 )
and

Λ s = t 2 + 3ν 2 (-ρ 2 1 + ρ 2 2 + 5ρ 2 3 -6ρ 1 ρ 2 + 12ρ 2 ρ 3 + 6ρ 3 ρ 1 ).
Denoting a = Λt-Λs 2 and b = Λt+Λs 2 , we diagonalize (2.3.4) by the symplectic change of variables 8

     ζ t -= 1 √ 1-α 2 (ζ s -iαζ t ) η t -= 1 √ 1-α 2 (η s -iαη t ) ζ t + = 1 √ 1-α 2 (ζ t + iαζ s ) η t + = 1 √ 1-α 2 (η t + iαη s ) where α = - a -a 2 -18 2 ν 4 ρ 2 1 ρ 2 ρ 3 ν 2 ρ 1 √ ρ 2 ρ 3 .
Then (2.3.4) becomes

Λ t + |ζ t + | 2 + Λ t -|ζ t -| 2 where Λ t ± = b ± a 2 -18 2 ν 4 ρ 2 1 ρ 2 ρ 3 .
We see that two modes t + , t -correspond to hyperbolic direction if and only if a 2 -18 2 ν 4 ρ 2 1 ρ 2 ρ 3 < 0, a condition related to the choice of ρ. Precisely, for ρ ∈ D 1 = [1, 2] 3 , we have Λ t ± ∈ R while for ρ = (2, 1, 9) we have a = 0 and

a 2 -18 2 ν 4 ρ 2 1 ρ 2 ρ 3 = -18 2 ν 4 ρ 2 1 ρ 2 ρ 3 < 0. Hence, there exist > 0(choose = 10 -2 ) such that for ρ ∈ D 2 = D = [2 -, 2 + ] × [1 -, 1 + ] × [9 -, 9 + ] we have |Im Λ t ± | > ν 2 .
We call Φ 2 the composition of changes of variables related to B.

For the set E, without loss of generality, assume that

     2p + q = m + 2s 2p 2 + q 2 = m 2 + 2s 2 .
(2.3.5)

Then, using the symplectic change of variables Ψ ang,2 :

O s (σ, µ) → O s (σ, 2µ); (r, θ, ζ) → (r , θ, ζ ) defined by 8. √ -1 = i                            ζ s = e iα/2 ζ s η s = e -iα/2 η s ζ j = ζ j , η j = η j j = s, p, q r 1 = r 1 + |ζ s | 2 r 2 = r 2 + 1 2 |ζ s | 2 r 3 = r 3 -1 2 |ζ s | 2 .
The effective part related to s becomes

Λ s |ζ s | 2 + ν 2 ρ 1 √ ρ 2 ρ 3 (ζ 2 s + η 2 s ) (2.3.6)
where

Λ s = 3ν 2 (2ρ 2 1 + ρ 2 2 -ρ 2 3 + 9ρ 1 ρ 2 + 3ρ 3 ρ 1 ). If Λ s = 0, we can diagonalize (2.3.6) into 1-β 2 1+β 2 Λ s | ζ s +βη s √ 1-β 2 | 2 with β satisfying Λ s β = (1 - β 2 )ν 2 ρ 1 √ ρ 2 ρ 3 , otherwise we rewrite it into iν 2 ρ 1 √ ρ 2 ρ 3 ( ζ s +iη s √ 2 η s +iζ s √
2 ), however meas{ρ ∈ R 3 : Λ s = 0} = 0. We call Φ 3 the composition of all changes of variables related to E.

By construction of Φ i and definition of O

s (σ, ν), the composition Φ 3 • Φ 2 • Φ 1 maps O s (σ, ν) into O s (σ, 3ν
). Using the rescaling Ψ introduced in (2.3.2), as the previous case we get Theorem 2.3.4. Assume that the equation (2.1.7) with j 1 , j 2 , j 3 ∈ {p, q, m} has no solution in Z and A, B, C, E are disjoint. The change of variables Φ

ρ := Ψ•Φ 3 •Φ 2 •Φ 1 •τ is a holomorphic, symplectic
1 ν (h • Φ ρ -C) = h 0 + f
where C is a constant and the effective part h 0 of the Hamiltonian reads

h 0 = Ω(ρ) • r + a∈Z Λ a |ζ a | 2
where

Ω(ρ) =      p 2 + 3ν 2 (ρ 2 1 + 3ρ 2 2 + 3ρ 2 3 + 6ρ 1 ρ 2 + 6ρ 1 ρ 3 + 12ρ 2 ρ 3 ) q 2 + 3ν 2 (ρ 2 2 + 3ρ 2 1 + 3ρ 2 3 + 6ρ 1 ρ 2 + 6ρ 2 ρ 3 + 12ρ 1 ρ 3 ) m 2 + 3ν 2 (ρ 2 3 + 3ρ 2 1 + 3ρ 2 2 + 6ρ 1 ρ 3 + 6ρ 3 ρ 2 + 12ρ 2 ρ 1 )     
-Z is the disjoint union L∪F; F is consistent with B and corresponds to hyperbolic part ; L is consistent with other exterior modes and corresponds to elliptic part ; -the remainder term f belongs to T s (σ, µ, D) and satisfies

[f ] s σ,µ,D ν 2 and [f T ] s σ,µ,D ν 7/2 .
Proof of Theorem 2.1.3. By Theorem 2.3.4, for (p, q, m) = (-3, 10, -6) and ρ ∈ D ν ⊂ D 2 , there exists a symplectic change of variables Φ ρ on D ν that puts the Hamiltonian h = N +P in normal form h 0 +f. In appendix A we verify that h 0 +f satisfies assumptions of KAM theorem 2.2.2 for δ = ν 2 , ε = ν 7/2 = δ 7/4 and Ω 0 = ω = (3 2 , 10 2 , 6 2 ) + O(ν 2 ). Since the hyperbolic set F is not empty, Φ -1 ρ • T lin ρ is an invariant KAM torus that is linearly unstable.

Appendix A

In this appendix, we will verify the hypothesis A0, A1, A2 of Theorem 2.2.2 for the Hamiltonian in our applications. The hypothesis A0, A1 is trivial, so we focus on A2.

Two-modes case

The first case In this case, we have F = ∅ and the other estimates are trivial. For the hypothesis A2, we recall that

Ω(ρ) =   p 2 + 3ν 2 (ρ 2 1 + 3ρ 2 2 + 6ρ 1 ρ 2 ) q 2 + 3ν 2 (ρ 2 2 + 3ρ 2 1 + 6ρ 1 ρ 2 )   and Λ j = j 2 + 9ν 2 ρ 2 1 + ρ 2 2 + 4ρ 1 ρ 2 .
Chapitre 2 -An unstable three dimensional KAM torus for the quintic NLS

Let k = (k 1 , k 2 ) ∈ Z 2 /{0} and z = z(k) = (k 2 ,k 1 )
|k| , then we have

(∇ ρ • z)(Ω(ρ) • k) = 6ν 2 3(ρ 1 + ρ 2 )k 2 2 + 3(ρ 2 + 3ρ 1 )k 2 1 + 4(ρ 1 + ρ 2 )k 1 k 2 |k| -1 ≥ 6 √ 2 ν 2 |k| and (∇ ρ • z)Λ j = 18ν 2 ((ρ 1 + 2ρ 2 )k 2 + (ρ 2 + 2ρ 1 )k 1 )|k| -1 .
Choosing δ = 4ν 2 , we get the hypothesis A2 (1). Since (∇ ρ • z)(Λ j -Λ ) = 0, the estimate of small divisor Ω • k + Λ j -Λ follows. To estimate the small divisors Ω • k + Λ j and Ω • k + Λ j + Λ we use the fact that f commutes with both the mass L and momentum M.

We just need to control small divisors Ω • k + Λ j and Ω • k + Λ j + Λ whenever e ik•θ η j ∈ f and e ik•θ η j η ∈ f , respectively. We have for the mass and momentum :

L = ν(ρ 1 + ρ 2 ) + r 1 + r 2 + j |ζ j | 2 and M = ν(pρ 1 + qρ 2 ) + pr 1 + qr 2 + j j|ζ j | 2 .
By conservation of L, we have

{e ik•θ η j , L} = ie ik•θ η j (k 1 + k 2 + 1) = 0.
Therefore, for A2 (2) we just have to study the case k 1 + k 2 = -1. In this situation

(∇ ρ • z)(Ω(ρ) • k + Λ j ) = 6ν 2 |k| -1 3(ρ 1 + ρ 2 )k 2 2 + 3(ρ 2 + ρ 1 )k 2 1 + 4(ρ 1 + ρ 2 )k 1 k 2 + 6ν 2 |k| -1 (3(ρ 1 + 2ρ 2 )k 2 + 3(ρ 2 + 2ρ 1 )k 1 ) = 6ν 2 |k| -1 (ρ 1 + ρ 2 )k 2 2 + (ρ 2 + ρ 1 )k 2 1 + 2(ρ 1 + ρ 2 ) + 6ν 2 |k| -1 (3ρ 2 k 2 + 3ρ 1 k 1 -3(ρ 1 + ρ 2 )) = 6ν 2 |k| -1 2(ρ 1 + ρ 2 )k 2 1 + (5ρ 1 -ρ 2 )k 1 -3ρ 2 .
This term is greater than δ except in the cases k = (-1, 0) and (0, -1). The conservation of M gives us

{e ik•θ η j , M} = ie ik•θ η j (pk 1 + qk 2 + j) = 0.
For k ∈ {(-1, 0), (0, -1)}, this implies j ∈ {p, q}, which is excluded. We consider the small divisor Ω • k + Λ j + Λ in the same way. The conservation of the mass L gives us k 1 + k 2 = -2 and then by computation we get k ∈ {(0, -2), (-2, 0), (-1, -1), (-3, 1), (1, -3)}.

The conservation of the momentum gives us pk 1 + qk 2 + j + = 0. We have

Ω • k + Λ j + Λ = N (p, q, j, ) + µ(ρ, k)
where N (p, q, j, ) = p 2 k 1 + q 2 k 2 + j 2 + 2 and µ(ρ, k) very small for |k| ≤ 4. We see that N (p, q, j, ) ∈ Z, so N (p, q, j, ) ≤ δ if and only if p 2 k 1 + q 2 k 2 + j 2 + 2 = 0. Combined with conservation of the momentum, this gives for the case k = (-1, -1)

p + q = j + and p 2 + q 2 = j 2 + 2
for the case k = (-2, 0) 2p = j + and 2p 2 = j 2 + 2 for the case k = (0, -2) 2q = j + and 2q 2 = j 2 + 2 for the case k = (-3, 1) 3p = q + j + and 3p 2 = q 2 + j 2 + 2 for the case k = (1, -3)

3q = p + j + and 3q 2 = p 2 + j 2 + 2 .
In all these cases, we get j, ∈ {p, q} which is excluded.

The second case We see that Ω and {Λ j } j =p,q,s,t are all the same as the previous case except Λ t + and Λ t -.We recall that

     2p + s = 2q + t 2p 2 + s 2 = 2q 2 + t 2 .
Thank to Lemma 2.2 in [START_REF] Grébert | Resonant dynamics for the quintic nonlinear Schrödinger equation[END_REF], {p, q, s, t} is in form of {p, p+2n, p+3n, p-n}. Without loss of generality, we can assume that 9 p = 0, so we have q = -2t. For Ω • k + Λ t + and Ω • k + Λ t -, by conservation the momentum, we just need to consider the case when k satisfies pk 1 + qk 2 + t = 0 i.e. k 2 = 1/2, which is not an integer. For Ω • k + Λ t ± ± Λ j , again by conservation of the momentum, we have

     pk 1 + qk 2 + t ± j = 0 p 2 k 1 + q 2 k 2 + t 2 ± j 2 = 0 i.e.      j = ∓(2k 2 -1)n j 2 = ∓(4k 2 + 1)n 2 .
This system has two solutions for j, either j = 0 = p or j = 3m = s, which are both excluded.

Three modes case.

It is too complicated to verify all the possibility, in this appendix we just consider the example (p, q, m) = (-3, 10, -6), which we are interesting in Theorem 2.1.3. In this situation, we have that C, E are empty, A = {-14, 2} and B = {9, 1}. Recall that

Ω(ρ) =      p 2 + 3ν 2 (ρ 2 1 + 3ρ 2 2 + 3ρ 2 3 + 6ρ 1 ρ 2 + 6ρ 1 ρ 3 + 12ρ 2 ρ 3 ) q 2 + 3ν 2 (ρ 2 2 + 3ρ 2 1 + 3ρ 2 3 + 6ρ 1 ρ 2 + 6ρ 2 ρ 3 + 12ρ 1 ρ 3 ) m 2 + 3ν 2 (ρ 2 3 + 3ρ 2 1 + 3ρ 2 2 + 6ρ 1 ρ 3 + 6ρ 3 ρ 2 + 12ρ 2 ρ 1 )      and Λ j = j 2 + 9ν 2 (ρ 2 1 + ρ 2 2 + ρ 2 3 + 4ρ 1 ρ 2 + 4ρ 2 ρ 3 + 4ρ 3 ρ 1 ) j = -14, -6, -3, 2, 1, 9, 10.
The hypothesis A0 and A1 are trivial. For hypothesis A2 (1

), let k = (k 1 , k 2 , k 3 ) ∈ Z 3 /{0}, k = (k 2 + k 3 , k 1 + k 3 , k 2 + k 1 ) and z = z(k) = k |k | , then we have (∇ ρ • z)(Ω(ρ) • k) =6ν 2 |k | -1 [ρ 1 (3k 2 2 + 3k 2 3 + k 1 k 2 + k 1 k 3 + 6(k 1 + k 2 + k 3 ) 2 ) + ρ 2 (3k 2 1 + 3k 2 3 + k 1 k 2 + k 2 k 3 + 6(k 1 + k 2 + k 3 ) 2 ) + ρ 3 (3k 2 2 + 3k 2 1 + k 3 k 2 + k 1 k 3 + 6(k 1 + k 2 + k 3 ) 2 )].
9. using the change of variables j = j -p

This term is greater than δ = ν 2 . Since (∇ ρ • z)(Λ j -Λ ) = 0, the estimate of small divisor Ω • k + Λ j -Λ follows.

For hypothesis A2 (2), (3), choose z = z(k) = -k |k| , then we have

(∇ ρ • z)(Ω(ρ) • k) = -6ν 2 |k| -1 [ρ 1 (k 2 1 + 3k 2 2 + 3k 2 3 + 6k 1 k 2 + 6k 1 k 3 + 12k 2 k 3 ) + ρ 2 (k 2 2 + 3k 2 1 + 3k 2 3 + 6k 1 k 2 + 6k 2 k 3 + 12k 1 k 3 ) + ρ 3 (k 2 3 + 3k 2 2 + 3k 2 1 + 6k 3 k 2 + 6k 1 k 3 + 12k 2 k 1 )]
and

(∇ ρ • z)Λ j = -18ν 2 |k| -1 [ρ 1 (k 1 + 2k 2 + 2k 3 ) + ρ 2 (k 2 + 2k 1 + 2k 3 ) + ρ 3 (k 3 + 2k 2 + 2k 1 )].
For Ω • k + Λ j , by conservation of the mass, we just need to estimate this divisor in the case k 1 + k 2 + k 3 = -1, then by computation we have

|(∇ ρ • z)(Ω(ρ) • k + Λ j ) = 6ν 2 |k| -1 [ρ 1 (2k 2 1 -6k 2 k 3 + 3k 1 + 3) + ρ 2 (2k 2 2 -6k 1 k 3 + 3k 2 + 3) + ρ 3 (2k 2 3 -6k 2 k 1 + 3k 3 + 3)] ≥ 6ν 2 |k| -1 [ρ 1 (2k 2 1 - 3 2 (k 1 + 1) 2 + 3k 1 + 3) + ρ 2 (2k 2 2 - 3 2 (k 2 + 1) 2 + 3k 2 + 3) + ρ 3 (2k 2 3 - 3 2 (k 3 + 1) 2 + 3k 3 + 3)] = 3ν 2 |k| -1 [ρ 1 (k 2 1 + 3) + ρ 2 (k 2 2 + 3) + ρ 3 (k 2 3 + 3)] ≥ ν 2 .
For Ω • k + Λ j + Λ , again we have k 1 + k 2 + k 3 = -2 by conservation of the mass, hence

|(∇ ρ • z)(Ω(ρ) • k + Λ j ) =6ν 2 |k| -1 [ρ 1 (2k 2 1 -6k 2 k 3 + 6k 1 + 12) + ρ 2 (2k 2 2 -6k 1 k 3 + 6k 2 + 12) + ρ 3 (2k 2 3 -6k 2 k 1 + 6k 3 + 12)] ≥ 6ν 2 |k| -1 [ρ 1 (2k 2 1 - 3 2 (k 1 + 1) 2 + 6k 1 + 12) + ρ 2 (2k 2 2 - 3 2 (k 2 + 2) 2 + 6k 2 + 12) + ρ 3 (2k 2 3 - 3 2 (k 3 + 2) 2 + 6k 3 + 12)] = 3ν 2 |k| -1 [ρ 1 (k 2 1 + 12) + ρ 2 (k 2 2 + 12) + ρ 3 (k 2 3 + 12)] ≥ ν 2 .
The set B For ρ ∈ D 2 : we have

|Im Λ 1 ± | > ν 2 = δ so that |Ω • k + Λ 1 + -Λ 1 -| ≥ 2ν 2 > δ.
For Ω • k + Λ 1 + + Λ 1 -, by conservation of the mass and the momentum, we just need to estimate this small divisor if

                   k 1 + k 2 + k 3 + 2 = 0 -3k 1 + 10k 2 -6k 3 + 2 = 0 9k 1 + 100k 2 + 36k 3 + 2 = 0 k 1 , k 2 , k 3 ∈ Z
This equation system has no solution 10 .

The set A For Ω • k + Λ 2 ± and Ω • k + Λ 2 ± + Λ j again the conservation of the mass and the momentum give

( * )            k 1 + k 2 + k 3 + 1 = 0 -3k 1 + 10k 2 -6k 3 + 2 = 0 9k 1 + 100k 2 + 36k 3 + 4 = 0 ( * * )            k 1 + k 2 + k 3 + 2 = 0 -3k 1 + 10k 2 -6k 3 + 2 + j = 0 9k 1 + 100k 2 + 36k 3 + 4 + j 2 = 0.
It is easy to see that ( * ) has no solution in Z 3 . For ( * * ) we have j ≡ -k 2 -2 (mod 3) and j 2 ≡ -k 2 -4 (mod 9). If j ≡ ±1 (mod 3) then we have k 2 ≡ 0, 2 (mod 4) and k 2 = 4 (mod 9), which can not both happen. If j ≡ 0 (mod 3) then we have k 2 ≡ 1 (mod 4) and k 2 = 5 (mod 9), which again can not happen. For Ω • k + Λ 2 ± -Λ j , because of changes of variables, we have

Λ 2 + = Λ 2 -g(ρ 1 , ρ 2 , ρ 3 ) Λ 2 -= Λ 2 -g(ρ 1 , ρ 2 , ρ 3 ) + 12(ρ 2 3 -ρ 2 2 + 3ρ 1 ρ 3 -3ρ 2 ρ 1 ) with g(x, y, z) = µ 2 81y 2 z 2 + (-18xy + 18xz -6y 2 + 6z 2 ) 2 -µ 2 (-18xy + 18xz -6y 2 + 6z 2 ).
10. with the implicit form of {p, q, m, s, t} in appendix B, we can solve for general p, q, m By conservation of the mass we just need to consider the case

k 1 + k 2 + k 3 = 0, then (∇ ρ • z)(Ω • k + Λ 2 ± -Λ j ) =12µ 2 |k| -1 [ρ 1 (k 2 2 + k 2 3 -k 2 k 3 -2k 2 + k 3 ) + ρ 2 (k 2 1 + k 2 3 -k 1 k 3 + 2k 1 -k 3 ) + ρ 3 (k 2 2 + k 2 1 -k 2 k 1 + 3k 1 -3k 2 )] ± (∇ ρ • z)g ≈ 12|k|µ 2 ρ ± |(∇ ρ • z)g|.
The conservation of the momentum implies

     -3k 1 + 10k 2 -6k 3 + 2 -j = 0 9k 1 + 100k 2 + 36k 3 + 4 -j 2 = 0.
The solution k of this equation system that closet to the origin is k = (-975, 195, 780) and with such a big k,

(∇ ρ • z)(Ω • k + Λ 2 ± -Λ j
) is far greater than δ.

Appendix B

In this appendix, we solve the set B in general

     2p + q = m + s + t 2p 2 + q 2 = m 2 + s 2 + t 2 . Let q 1 = q -p, m 1 = m -p, s 1 = s -p, t 1 = t -p, it becomes      q 1 = m 1 + s 1 + t 1 q 2 1 = m 2 1 + s 2 1 + t 2 1 .
This give us m 1 s 1 + t 1 s 1 + t 1 m 1 = 0, hence s 1 = -m 1 t 1 m 1 +t 1 . Assume more that s 1 , t 1 , m 1 have no common divisor except ±1. Let k is a prime common divisor of t 1 and m 1 , i.e.

t 1 = t 2 k, m 1 = m 2 k, then s 1 = -km 2 t 2 m 2 +t 2 . Since k s 1 , we have k | t 2 + m 2 , i.e. t 2 = kh -m 2 , hence s 1 = -m 2 (kh-m 2 ) h = -km 2 + m 2 2 h ∈ Z. Let h = (-1) sgn(h) Πp k i i , x = Πp [ k i 2 ] i and y = (-1) sgn(h) Πp k i -2[ k i 2 ] i
, with p i is prime divisor of h. Then, h = x 2 y and we need xy | m ťť , i.e. m ťť = ryx. By this, s 1 = -kxyr + r 2 y, m 1 = kryx, t 1 = k 2 x 2 y -ryx. Since s 1 , t 1 , m 1 have no common divisor except ±1, we have y = ±1. Assume that y = 1, and kx = n, then s 1 = r 2 -nr, m 1 = nr, t 1 = n 2 -nr and q 1 = n 2 -nr + r 2 . In general, we have {p, q, m, s, t} = {p, p + k(n 2 -nr + r 2 ), p + knr, p + k(r 2 -nr), p + k(n 2 -nr)}.

3

Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential

Introduction

The reducibility problem for Schrödinger equations with quasi-periodic in time perturbation has been intensively studied in recent years. The first results adapting the KAM technics were due to Kuksin [START_REF] Kuksin | Nearly integrable infinite-dimensional Hamiltonian systems, sous la dir[END_REF] followed by many results in one dimensional context (see in particular [BG01 ; LY10 ; GT11]). More recently the techniques were adapted to the higher dimensional case [EK09 ; EGK16 ; GP19]. To consider unbounded perturbations, a new strategy has been developed in [BBM14 ; BBM16] using the pseudo-differential calculus. Without trying to be exhaustive we quote also [FGP ; FP14 ; BM20 ; BBHM18] regarding KAM theory for quasi-linear PDEs in one space dimension. This technics were successfully applied for reducibility problems in various cases. For one dimensional linear equations with unbounded potential we quote [Bam17 ; BM18 ; FGP18]. In higher space dimensions with unbounded perturbations only few results exist, one concerning the quantum harmonic oscillator on R n with polynomial time dependent perturbation [START_REF] Bambusi | Reducibility of the quantum harmonic oscillator in d-dimensions with polynomial time-dependent perturbation[END_REF] and some special examples on the torus T n [Mon19 ; FGMP19 ; BLM19]. In all these multi-dimensional examples the unperturbed linear system were integrable in the classical sense (for instance on the torus T n , the Laplacian operator commutes with

∂ j , j = 1, • • • , n),
a fact that will be crucial in the control of the perturbed spectrum (see (3.1.11) below). In this article we consider a Schrödinger equation on a Zoll manifold on which the Laplace Beltrami operator ∆ g has, in general, no other first integral than energy (in particular ∆ g doesn't commute with ∂ j ) .

We first recall that a Zoll manifold of dimension n ∈ N is a compact Riemannian manifold (M n , g) such that all the geodesic curves have all the same period T . In this paper we assume T := 2π. For example the n-dimensional sphere S n is a Zoll mani-Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential fold. We denote by ∆ g the positive Laplace-Beltrami operator on (M n , g) and we define H s (M n ) := dom( 1 + ∆ g ) s with s ∈ R the usual scale of Sobolev spaces. We denote by S m cl (M n ) the space of classical real valued symbols of order m ∈ R on the cotangent bundle T * (M n ) and we define A m the associated class of pseudo-differential operators (see for instance Hörmander [START_REF] Hörmander | The analysis of linear partial differential operators I-III, sous la dir[END_REF] for a definition of pseudo-differential operators on a manifold see also [START_REF] Bambusi | Growth of Sobolev norms for abstract linear Schrödinger equations[END_REF] in the case of a Zoll manifold).

We consider the following linear Schrödinger equation

i∂ t u = ∆ g u + εW (ωt)u , u = u(t, x) , t ∈ R , x ∈ M n , (3.1.1)
where ε > 0 is a small parameter and W (ωt) is a time dependent unbounded operator from The purpose of this article is to construct a change of variables that transforms the non-autonomous equation (3.1.1) into an autonomous equation.

H s (M n ) → H s-δ (M n ) for some δ ≤ 1/2. More precisely we assume that W ∈ C ∞ (T d ; A δ ) with δ ≤ 1/2,
Our main result is the following. Theorem 3.1.1. Let 0 < α < 1 and δ ∈ R, δ ≤ 1/2. Assume that the map ϕ → W (ϕ, •) ∈ A δ is C ∞ in ϕ ∈ T d . Then for any s ∈ R, s > n/2 there exists ε 0 > 0 and C > 0 such that, for any 0 < ε ≤ ε 0 there is a set

O ε ⊂ [1/2, 3/2] d ⊂ R d with meas([1/2, 3/2] d \ O ε ) ≤ Cε α (3.1.2)
such that the following holds. For any ω ∈ O ε there exists a family of linear isomorphisms Ψ(ϕ) ∈ L(H s (M n )) and a Hermitian operator Z ∈ A δ commuting with the Laplacian1 and satisfying

Z L(H s (M n ),H s-δ (M n )) ≤ Cε . (3.1.3) Furthermore • Ψ(ϕ) is unitary on L 2 (M n ) ; 3.1. Introduction • for any n 2 < s ≤ s and any ω ∈ O ε Ψ(ϕ) -Id L(H s (M n ),H s -δ (M n )) + Ψ(ϕ) -1 -Id L(H s (M n ),H s -δ (M n )) ≤ Cε 1-α , Ψ(ϕ) L(H s (M n )) + Ψ(ϕ) -1 L(H s (M n )) ≤ 1 + Cε 1-α , ( 3
i∂ t v = ∆ g v + εZ(v) .
(3.1.5)

As a consequence of our reducibility result, we get a control of the flow generated by the (3.1.1) equation in the scale of Sobolev spaces :

Corollary 3.1.2. Let W ∈ C ∞ (T d ; A δ ) with δ ≤ 1/2. Then for any s ∈ R, s > n/2 there exists ε 0 > 0 and C > 0 such that, for any 0 < ε ≤ ε 0 there is a set O ε ⊂ [1/2, 3/2] d ⊂ R d satisfying (3.1.2) such that for any ω ∈ O ε the flow generated by the (3.1.1) equation is bounded in H s (M n ). More precisely if u 0 ∈ H s (M n ) then there exists a unique solution u ∈ C 1 R ; H s (M n ) of (3.1.1
) such that u(0) = u 0 . Moreover, u is almost-periodic in time and satisfies

(1 -εC) u 0 H s ≤ u(t) H s ≤ (1 + εC) u 0 H s , ∀ t ∈ R , (3.1.6)
for some C = C(s) > 0.

Following the pioneering work [START_REF] Baldi | KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation[END_REF] we prove Theorem 3.1.1 in two steps :

-The regularization step where we use the pseudo-differential calculus (and in particular the technics developed in [START_REF] Bambusi | Growth of Sobolev norms for abstract linear Schrödinger equations[END_REF]) to transform equation (3.1.1) in a system with a smoothing perturbation, still depending on time ; -The KAM step where we use a KAM procedure (going back to [START_REF] Kuksin | Nearly integrable infinite-dimensional Hamiltonian systems, sous la dir[END_REF] but using recent development in [START_REF] Baldi | Time quasi-periodic gravity water waves in finite depth[END_REF]) on infinite dimensional matrices to eliminate the time in the new system.

The same strategy was recently successfully applied in [START_REF] Bambusi | Reducibility of Non-Resonant Transport Equation on with Unbounded Perturbations[END_REF] to prove the reducibility of non-resonant transport equation on the torus T n . Our main contribution consists in merging these two recent technics in the context of linear Schrödinger equation on Zoll manifold which, in contrast to the transport equation on the torus, is not an integrable system.

Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential

Our result has to be compared with the recent work of two of us [START_REF] Feola | Reducibility of Schrödinger equation on the sphere[END_REF] where we consider a Schrödinger equation on the sphere S n with a quasi-periodic in time odd perturbations of order < 1/2. In that case a more standard approach following [START_REF] Grébert | KAM for KG on the sphere Sd[END_REF] was possible, in particular our analysis did not require the pseudo-differential calculus. Of course this new paper is a generalization in the sense that we replaced the sphere by a Zoll manifold and we are able to treat perturbations of order 1/2. But we want to stress that the elimination of the potential symmetry hypothesis may be even more important if we look at generalization to the non-linear case. Actually a natural strategy to solve the non linear Schrödinger eqation

i∂ t u = ∆ g u + mu + ε|u| 2 u , u = u(t, x) , t ∈ R , x ∈ M n , (3.1.7)
consists in a Newton scheme : we linearize the equation (3.1.7) around an approximate solution u 0 , we solve this linear equation to obtain u 1 and we linearize (3.1.7) around u 1 and we iterate. Doing so we have to solve linear Schrödinger equation of the kind2 (3.1.1) where W (ωt) = |u 0 (t)| 2 which is clearly not an odd function.

As a matter of fact the existence of quasi-periodic solutions of the forced non linear Schrödinger equation

i∂ t u = ∆ g u + mu = εf (ωt, x, u) , u = u(t, x) , t ∈ R , x ∈ S n (3.1.8)
were already addressed by Berti-Corsi-Procesi in [START_REF] Berti | An Abstract Nash Moser Theorem and Quasi-Periodic Solutions for NLW and NLS on Compact Lie Groups and Homogeneous Manifolds[END_REF]. They proved that for ω in a large Cantor's set and for a Hamiltonian and smooth forced nonlinear perturbation f and ε small enough, there is a smooth quasi-periodic solution of (3.1.8). See also [START_REF] Berti | Nonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces[END_REF] by Berti-Bolle-Procesi in which the authors prove existence of periodic solutions on Zoll manifolds, we also mention [START_REF] Corsi | A KAM result on Compact Lie Groups[END_REF] where a KAM approach was considered in the context of Lie groups with symmetries. An adaptation of our work to the context of systems of linear Schrödinger equations (see previous footnote) could prove that this solution is linearly stable. We shall remark that such an adaptation is not trivial at all, and some new ideas are required.

We list here the main issues :

• as already said the linearization of (3.1.8) give rise to a linear operators acting on the couple u u , i.e. and equation of the form i∂ t u = ∆ g u + εW 1 (ωt)u + εW 2 (ωt)u , (3.1.9) with W i , i = 1, 2 operators similar to W in (3.1.1). As first step of the regularization procedure, one needs to block diagonalize, up to smoothing remainders, the operator in the r.h.s. of (3.1.9) in the spirit of [START_REF] Feola | Local well-posedness for quasi-linear NLS with large Cauchy data on the circle[END_REF], [START_REF] Feola | Long time existence for fully nonlinear NLS with small Cauchy data on the circle[END_REF], i.e. one needs to "eliminate" W 2 . This is just a technical point which can be addressed by extending the arguments of section 3.3.

• A second difficulty concerns the pseudo-differential calculus on Zoll manifold which is more difficult and more implicit than the pseudo-differential calculus on the torus.

In particular the estimates on the semi-norms of the symbols (see the estimates under Definition 3.2.1) are not sharp in terms of the regularity required on symbols. Actually on tori one can define, explicitly a semi-norm satisfying "tame" estimates. We refer, for instance, to section 2 in [START_REF] Feola | Reducibility for a class of weakly dispersive linear operators arising from the Degasperis Procesi equation[END_REF].

• A major difficulty (which is linked to the item above) regards the minimal regularity one needs on the potential W (ωt) in (3.1.1) (or W 1 , W 2 in (3.1.9)). Theorem 3.3.1 shows that, in order to prove reducibility in H s (M n ), s > n/2, the potential W (ωt) must be in H p , both in time and space, for some p 1 depending on s. This is a consequence of the use of pseudo-differential calculus for the regularization. In the proof of Theorem 3.1.1 in section 3.5 it turns out that p 2s. In the case of this paper this is not a problem since the potential is C ∞ . In the non linear case, the regularity of the potential depends on the approximate solution of the previous step and hence have only finite regularity. Moreover the requirement p 2s is not compatible with the convergence of the Nash-Moser scheme. We notice that in 1d this problem can be overcome (see for instance [START_REF] Baldi | Time quasi-periodic gravity water waves in finite depth[END_REF] or [START_REF] Feola | Reducibility for a class of weakly dispersive linear operators arising from the Degasperis Procesi equation[END_REF]) since we need only few regularization steps and furthermore we can use the so called Poschel's Lemma (see Lemma A.1 in [START_REF] Pöschel | A KAM-theorem for some nonlinear partial differential equations[END_REF]). It is not clear at the moment how to overcome this problem in a multi-dimensionnal context.

• Another serious problem is about the small divisors. It is know that a KAM reducibility scheme requires some non resonance condition on the eigenvalues of the linear operator. We prove such non resonance conditions, for many frequency vector ω, in section 3.4.1. It turns out that the eigenvalues of the operator in the right hand side of (3.1.1) have the expansion

µ k,j = Λ k,j + µ (1) k,j + µ (2) k,j , k ∈ N , j = 1, . . . , d k ,
where Λ k,j are the eigenvalues of ∆ g (see (3.2.5)), d k the dimension of the corresponding eigenspace, and

µ (1) k,j ∼ ε|k| δ , µ (2) k,j ∼ ε ,

Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential

with δ the order of the pseudo-differential operator W (ωt). In the measure estimate Lemma 3.4.3 it is fundamental that the unbounded corrections µ

(1) k,j do not depend on the parameters ω ; as a consequence the Lipschitz norm of the eigenvalues µ k,j is bounded in k. In our case this fact is true since the corrections µ

(1) k,j are obtained form an averaging procedure on the potential W , which, at the beginning, does not depend explicitly on ω. More precisely it depends on ω only through the variable ϕ = ωt. In the non linear case this is no more true, since W would depend on ω also through the function on which we linearized the equation (3.1.8). This problem could be overcome by considering non linear equation like (3.1.8) with bounded non linearity, in order to obtain a linear bounded operator when linearizing (i.e. δ ≤ 0).

In view of the issues discussed above the KAM result for a non linear Schrödinger equation with unbounded non linearities on Zoll manifold is out of reach for the moment. Nevertheless we believe that this paper represent an important milestone in that direction.

Scheme of the proof.

As said above the proof consists in a regularization step (section 3.3) and a KAM step (section 3.4). In section 3.5 we merge the two procedures to prove Theorem 3.1.1.

In the regularization step we prove that we can transform (by using a symplectic map :

u = Φ(v)) the original Schrödinger equation (3.1.1) in a new one i∂ t v = ∆ g v + ε(Z + R(ωt))v , (3.1.10)
where Z is a pseudo-differential operator of order δ independent on time and commuting with ∆ g and R is a ρ-regularizing operator in L(H s (M n ), H s+ρ (M n )) with ρ arbitrary large.

It is based on a normal form procedure developed in [START_REF] Bambusi | Growth of Sobolev norms for abstract linear Schrödinger equations[END_REF]. The crucial fact is that we can write ∆ g = K 0 -Q where Q is a pseudo-differential operator of order -1 chosen (following [START_REF] De Verdière | Sur le spectre des opérateurs elliptiques à bicaractéristiques toutes périodiques[END_REF]) in such a way the spectrum of K 0 is included in N+λ for some constant λ ∈ R + . This property makes the K 0 flow periodic and motivates us to use it to average the original Schrödinger operator : if A is a pseudo-differential operator then its average with respect to the flow of K 0 is given by A = 2π 0 e -iτ K 0 Ae iτ K 0 dτ . This idea was already used in a pioneering work of Weinstein [START_REF] Weinstein | Asymptotics of eigenvalue clusters for Laplacian plus a potential[END_REF].

Let us sketch the procedure. Let us write H = H 0 + V (t) where H 0 = ∆ g and V (t) = εW (ωt) is a pseudo-differential operator of order δ (for this averaging procedure we do not need to assume that V depends quasi-periodically on time neither than V is small). We conjugate the flow of H by a Lie transform e iX(t) where X(t) is a pseudo-differential operator of order δ -1 : if i u = H(t)u, in the new variable v = e iX(t) u we get i v = H + (t)v with (see subsection 3.2.4)

H + = H 0 + i[X, H 0 ] + V + i[X, V ] - 1 2 [X, [X, H 0 ]] + R .
Using the pseudo-differential calculus we get that i[X, V ] -1 2 [X, [X, H 0 ]] is of order 2δ -2 and that the remainder term R is still of higher order. So if we are able to solve the following homological equation (and we show in Lemma 3.3.2 how to do it) i[X, H 0 ] + V = V + order δ -1 , we conclude that H + = H 0 + V (t) + order δ -ν with ν = min(1, 2 -δ). Thus if δ < 2 we have a better equation. In [START_REF] Bambusi | Growth of Sobolev norms for abstract linear Schrödinger equations[END_REF] such a procedure was iterated to obtain an equivalent equation like (3.1.10) but with Z still depending on time and this was used to prove that we can control the Sobolev norms of the solutions of (3.1.1) as follows3

∀s > n/2 , ∀ν > 0 , ∃ C s,ν such that u(t) H s ≤ C s,ν (1 + t) ν .
In this paper, we want more : we want to eliminate totally the time in order to obtain (3.1.6). So we alternate the averaging procedure with a time elimination procedure based on the use of the operator (3.3.27) which solves the homological equation (3.3.28) and thus the Lie transform Φ T = e iT will kill the dependence on time in Z = V (see Lemma 3.3.4). This time elimination procedure requires a non resonance hypothesis on the frequency vector ω (see (3.3.2)) and requires δ < 1.

Throughout section 3.3 we work at the pseudo-differential level and the main difficulty is to precisely control the flow generated by pseudo-differential operator of positive order (see Appendix 3.6.3 and in particular hypothesis (3.6.13)). We notice that all this section holds true upon the hypothesis δ < 1.

In the KAM step we kill the remainder term R in (3.1.10) which still depends on time but is now a regularizing operator. As in [START_REF] Baldi | Time quasi-periodic gravity water waves in finite depth[END_REF] (see also [START_REF] Montalto | A reducibility result for a class of linear wave equations on[END_REF] and [START_REF] Bambusi | Reducibility of Non-Resonant Transport Equation on with Unbounded Perturbations[END_REF]) we use a reducibility scheme where the regularizing property of the perturbation compensates the bad non resonance estimates satisfied by the eigenvalues of ∆ g +εZ (see (3.4.13)). The condition δ ≤ 1/2 is used to ensure that condition (3.4.13) is preserved during the KAM iteration as long as a small part of the parameters ω are excised (see Lemma (3.4.3) where Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential κ = 1 -2δ). This constraint in the KAM procedure was not necessary in [START_REF] Bambusi | Reducibility of Non-Resonant Transport Equation on with Unbounded Perturbations[END_REF] (they obtain the reducibility for perturbation of order 1e for any e > 0 when the transport operator is of order 1) essentially because the unperturbed system is integrable : in the context of the transport equation on T n , H 0 = ν • ∇ with ν ∈ R n and thus H 0 commutes with ∂ m , m = 1, • • • , n and the same is true for H 0 + Z obtained after the regularization procedure. So Z is not depending on x and H 0 + Z is still diagonal in Fourier variables. Thus the perturbed eigenvalues have the form,

λ j = λ (0) j + z(j) + remainder , (3.1.11)
where z is the symbol of Z (see formula (4.13) in [START_REF] Bambusi | Reducibility of Non-Resonant Transport Equation on with Unbounded Perturbations[END_REF]). In our case we just know that Z commutes with ∆ g and thus we can just prove that the spectrum of ∆ g + V preserves the cluster structure inherited from ∆ g on a Zoll manifold. That means that, once written in the laplacian diagonalization basis, the matrix of Z is block-diagonal but not diagonal as in [START_REF] Bambusi | Reducibility of Non-Resonant Transport Equation on with Unbounded Perturbations[END_REF]. By the way throughout section 3.4 we work at the matrix level.

As usual the homological equation (3.4.16) is solved blockwise and it is well known that the increasing size of the blocks may generate loss of regularity. In [START_REF] Hakan | KAM for the nonlinear Schrödinger equation[END_REF] Eliasson-Kuksin used geometrical arguments (related to a Bourgain's Lemma, see Lemma 8.1 in [START_REF] Bourgain | Global wellposedness of defocusing critical nonlinear Schrodinger equation in the radial case[END_REF]) to control the size of the blocks, in [START_REF] Grébert | KAM for KG on the sphere Sd[END_REF] or [START_REF] Feola | Reducibility of Schrödinger equation on the sphere[END_REF] authors used a different argument introduced by Delort-Szeftel in [START_REF] Delort | Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres[END_REF] (see Lemma 4.3 in [START_REF] Grébert | KAM for KG on the sphere Sd[END_REF]). In this paper, as a consequence of the regularization step, we can solve the homological equation with loss of regularity and thus this step is simplified.

On the other hand the KAM procedure of [START_REF] Baldi | Time quasi-periodic gravity water waves in finite depth[END_REF] requires a tame property to deal with product of matrices. This motivates the definition of the space M s of matrices with s-decay norm (see Definition 3.2.8) which was first introduced in [BCP15] (see also [START_REF] Berti | Nonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces[END_REF]). The tame property for the s-decay norm is stated in Lemma 3. Section 3.3 and section 3.4 are independent and in fact are at different levels : while all section 3.3 takes place in the context of pseudo-differential operators, all section 3.4 takes place at matrix level. In section 3.5 we merge the two sections and for that we need the Lemma 3.2.14 which makes the link between ρ-smoothing operators and β-regularizing matrices.

Functional setting

In this section we introduce the space of functions, sequences, linear operators and pseudo differential operators we shall use along the paper.

Spectral decomposition

Following Theorem 1 of Colin de Verdière [START_REF] De Verdière | Sur le spectre des opérateurs elliptiques à bicaractéristiques toutes périodiques[END_REF], we introduce Q the pseudo-differential operator of order -1, commuting with ∆ g such that, setting

K 0 := ∆ g + Q , (3.2.1)
we have spec(K 0 ) ⊂ N + λ for some constant λ ∈ R + . We notice that our original Schrödinger operator H(t) := ∆ g + εW (ωt) reads

H(t) = ∆ g + εW (ωt) = K 2 0 + Q 0 + εW (ωt) (3.2.2)
where Q 0 = -2Q ∆ g -Q 2 is a pseudo differential operators of order 0.

Let us denote by λ k the eigenvalue of K 0 and by E k be the eigenspace associated to λ k . We have

λ k ∼ k , dimE k := d k ≤ k n-1 . (3.2.3)
We denote by

Φ [k] (x) := {Φ k,m (x) , m = 1, . . . , d k } (3.2.4)
an orthonormal basis of E k . By formula (3.2.1) we also deduce that ∆ g := K 2 0 + Q 0 where Q 0 is a pseudo differential operator commuting both with the Laplacian ∆ g and K 0 . For this reason K 0 and ∆ g diagonalize simultaneously, hence

∆ g Φ k,j = Λ k,j Φ k,j , k ∈ N , j = 1, . . . , d k , (3.2.5)

Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential

with

Λ k,j = λ 2 k + η k,j , |η k,j | 1 .
In particular there exists c 0 > 0 such that

Λ k,j ≥ c 0 k 2 , |Λ k,j -Λ k ,j | ≥ c 0 (k + k ) , ∀ k = k , (3.2.6)
and for any j = 1, . . . , d k , j = 1, . . . , d k .

Space of functions and sequences

Using the spectral decomposition of the space

L 2 (M n ) = ⊕ k∈N E k , any function u ∈ L 2 (M n ) can be written as u(x) = k∈N d k m=1 z k,m Φ k,m (x) = k∈N z [k] • Φ [k] (x) , z [k] = (z k,1 , • • • , z k,d k ) ∈ C d k , (3.2.7)
where • denotes the usual scalar product in R d k . We denote by Π E k the L 2 -projector on the eigenspace E k , i.e., for any k ∈ N,

(Π E k u)(x) = z [k] • Φ [k] (x) ⇒ ( √ -∆ + Q)Π E k u = λ k Π E k u . (3.2.8)
For s ≥ 0, we define the (Sobolev) scale of Hilbert sequence spaces

h s := z ={z [k] } k∈N , z [k] ∈ C d k : z 2 hs := k∈N k 2s z [k] 2 < +∞ , (3.2.9)
where k := 1 + |k| 2 and • denotes the L 2 (C d k )-norm. By a slight abuse of notation we define the operator Π E k on sequences as

Π E k z = z [k] for any z ∈ h s and k ∈ N.
We notice that the weight k we use in the norm in (3.2.9) is related to the eigenvalues of K 0 , indeed

c|k| ≤ λ k ≤ C|k| (3.2.10)
for some suitable constants 0 < c ≤ C. As a consequence the space

H s = H s (M n ) := {u(x) = k∈N z [k] • Φ [k] (x) | z ∈ h s } , (3.2.11) is the usual Sobolev space H s = dom((K 0 ) s ) = dom( 1 + ∆ g ) s and u H s := z hs is equivalent to the standard Sobolev norm u H s ∼ K s 0 u L 2 (M n ) .
Along the paper we shall write • H s instead of • hs . Given s, s ∈ R we denote by L(H s , H s ) the space of linear bounded operators form H s to H s endowed with the standard operator norm • L(H s ,H s ) .

In the paper we shall also deal with quasi periodic in time functions R × M n (t, x) → u(ωt, x) where ω ∈ R d is a frequency vector and u is periodic in its first variable. To this end we introduce the space H r (T d ; H s (M n )) defined as the set of functions u :

T d ϕ → H s (M n ) which are Sobolev in ϕ ∈ T d with values in H s (M n ).
Functions in H r (T d ; H s (M n )) can be expanded, using the standard Fourier theory, as

u(ϕ, x) = ∈Z d ,k∈N z [k] (l) • Φ [k] (x)e il•ϕ , z [k] (l) ∈ C d k , (3.2.12)
where

e il•ϕ Φ k,m (x), l ∈ Z d , k ∈ N, m = 1, . . . , d k , is an orthogonal basis of L 2 (T d × M n ; C).
We define space of sequence (recall (3.2.9))

h s,r := z = {z [k] (l)} l∈Z d ,k∈N , z [k] ∈ C d k : z 2 hs,r := l∈Z d l 2r z(l) 2 hs < +∞ . (3.2.13)
Along the paper we shall also consider the space, for p ∈ N with p > (d + n)/2, p := r>d/2,s>n/2 s+r=p h s,r .

(3.2.14)

We endow the space p with the norm

z 2 p := l∈Z d ,k∈N l, k 2p z [k] (l) 2 . (3.2.15) Lipschitz norm. Consider a compact subset O of R d , d ≥ 1. For functions f : O → E,
with (E, • E ) some Banach space, we define the sup norm and the lipschitz semi-norm as

f sup E := f sup,O E := sup ω∈O f (ω) E , f lip E := f lip,O E := sup ω 1 ,ω 2 ∈O ω 1 =ω 2 f (ω 1 ) -f (ω 2 ) E |ω 1 -ω 2 | . (3.2.16)
For any γ > 0 we introduce the weighted Lipschitz norms

f γ,O E := f sup,O E + γ f lip,O E .
(3.2.17)

Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential

We finally define the space of sequences Notation. We shall use the notation A B to denote A ≤ CB where C is a positive constant depending on parameters fixed once for all : d, n, δ. We shall use the notation A ≤ s B to denote A ≤ C(s)B where C(s) > 0 is a constant depending also on s.

h γ,O s,r := O ω → z(ω) ∈ h s,r : z γ,

Pseudo-differential operators

In this paper we consider operators which are pseudo-differential. Here we recall some fundamental properties of operators in A m which are collected in [START_REF] Bambusi | Growth of Sobolev norms for abstract linear Schrödinger equations[END_REF]. First A m is a Fréchet space for a family of filtering semi-norms {N m,p } p≥1 such that the embedding (recall (3.2.11)) A m → s∈R L(H s , H s-m ) is continuous. We can also choose the seminorms in an increasing way, i.e. N m,p (A) ≤ N m,p+1 (A) for p ≥ 1 and A ∈ A m . To state the other properties we need to introduce the following definition. Definition 3.2.1. Let S ∈ L(H 0 ). We say that S is ρ-smoothing, and we will write S ∈ R ρ , if S can be extended to an operator in L(H s , H s+ρ ) for any s ∈ R. When this is true for every ρ ≥ 0, we say that S is a smoothing operator.

Then we have the following properties concerning the class A m equipped with the seminorms {N m,p } p≥1 :

(i) let A ∈ A m , for any s ∈ R there exist constants C = C(m, s) > 0, p = p(m, s) ≥ 1
which are increasing functions4 of s such that

A L(H s ,H s-m ) ≤ CN m,p (A) . (3.2.21) 3.2. Functional setting (ii) Let A ∈ A m , B ∈ A n then AB ∈ A m+n .
Furthermore for any ρ ≥ 0 there exists S a ρ-smoothing operator such that for any p ≥ 1 for any s ∈ R there are constants

C = C(m, n, p, s, ρ) > 0, q = q(m, n, p, s, ρ) ≥ p such that N m+n,p (AB -S) ≤ CN m,q (A)N n,q (B) , (3.2.22) S L(H s ,H s+ρ ) ≤ CN m,q (A)N n,q (B) . (3.2.23) (iii) Let A ∈ A m , B ∈ A n then [A, B] ∈ A m+n-1
. Furthermore for any ρ ≥ 0 there exists S a ρ-smoothing operator such that for any p ≥ 1 for any s ∈ R there are constants C = C(m, n, p, s, ρ) > 0, q = q(m, n, p, s, ρ) ≥ p such that

N m+n-1,p ([A, B] -S) ≤ CN m,q (A)N n,q (B) , (3.2.24) S L(H s ,H s+ρ ) ≤ CN m,q (A)N n,q (B) . (3.2.25) (iv) The map τ → A(τ ) := e -iτ K 0 Ae iτ K 0 ∈ C 0 b (R, A m )
. Furthermore for any ρ ≥ 0 there exists S a ρ-smoothing operator such that for any p ≥ 1 for any s ∈ R there are constants C = C(m, n, p, s, ρ) > 0, q = q(m, n, p, s, ρ) ≥ p such that N m+n-1,p (e -iτ K 0 Ae iτ K 0 -S) ≤ CN m,q (A) , (3.2.26)

S L(H s ,H s+ρ ) ≤ CN m,q (A) . (3.2.27)
Remark 3.2.2. In (ii), (iii) and (iv) the smoothing correction does not play an important role since it can be chosen as regularizing as one want. In the KAM scheme the level of regularization will be fix once for all. Thus, by a slight abuse of notation, we will often omit in the following the smoothing correction and will just write

N m+n,p (AB) ≤ CN m,q (A)N n,q (B) , (3.2.28) N m+n-1,p ([A, B]) ≤ CN m,q (A)N n,q (B) . (3.2.29)
We shall also consider H r -mappings T d ϕ → A(ϕ) with A(ϕ) a symmetric pseudodifferential operators of order m in A m . We can then decompose A in Fourier writing

A(ϕ) = l∈Z d A(l)e il•ϕ
(3.2.30)

Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential

with A(l) a pseudo-differential operators of order m in A m . We give the following definition.

Definition 3.2.3. Let m ∈ R, r > d/2. We denote by A m,s the Fréchet space of mapping

T d ϕ → A = A(ϕ) ∈ A m that are H r on T d . We endow A m,
Ah h γ,O s-m,r ≤ CN γ,O m,r,p (A) h h γ,O s,
Rh h s+ρ,r ≤ l∈Z d l 2r l ∈Z d R(l -l )h(l ) H s+ρ 2 ≤ l∈Z d l 2r l ∈Z d R(l -l ) L(H s ;H s+ρ ) h(l ) H s 2 ≤ l∈Z d |l |> 1 2 |l| l -l r R(l -l ) L(H s ;H s+ρ ) l r h(l ) H s l r l r 2 + l∈Z d |l |≤ 1 2 |l| l -l r R(l -l ) L(H s ;H s+ρ ) l r h(l ) H s l r l -l r 2 .
Hence, by using the Cauchy-Schwartz inequality, we get In the following Lemma we state some properties and estimates 5 that will be proved in Appendix 3.6.1. A γ,O m,r and A γ,O n,r . For any p ≥ 1 there exist constants C = C(r, m, n, p) and q = q(r, m, n, p) which are increasing in p such that ) should be interpreted as : for any ρ ≥ 0 there exists S a ρ-smoothing operator such that for any p ≥ 1 for any s ∈ R there are constants C = C(m, n, p, r, ρ) > 0, q = q(m, n, p, r, ρ) ≥ 1 such that

Rh h s+ρ,r ≤ C l,l ∈Z d l -l 2r R(l -l ) 2 L(H s ;H s+ρ ) l 2r h(l ) 2 H s ≤ C h 2 hs,r |R| 2 ρ,

Lemma 3.2.6. Let A, B are pseudo-differential operators in

(i) AB, BA ∈ A γ,O m+n,r and N γ,O m+n,r,p (AB) , N γ,O m+n,r,p (BA) ≤ CN γ,O m,r,q (A)N γ,O n,r,q (B) . (3.2.37) (ii) [A, B] ∈ A γ,O m+n-1,r and N γ,O m+n-1,r,p ([A, B]) ≤ CN γ,O m,r,q (A)N γ,O n,r,q (B) . ( 3
N γ,O m+n-1,r,p ([A, B] -S) ≤ CN γ,O m,r,q (A)N γ,O n,r,q (B) , |S| γ,O ρ,s,p ≤ CN γ,O m,r,q (A)N γ,O n,r,q (B) .
Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential

(iii) Let ω ∈ R d , then ω • ∂ ϕ A ∈ A m,r-1 and N γ,O m,r-1,p (ω • ∂ ϕ A) ≤ CN γ,O m,r,p (A) . (3.2.39) If furthermore ω satisfies, for some α > d -1, |ω • l| > γ|l| -α , ∀ l ∈ Z d \ {0} , (3.2.40) and r -2α -1 > d/2 then (ω • ∂ ϕ ) -1 A ∈ A m,r-(2α+1) and N γ,O m,r-(2α+1),p ((ω • ∂ ϕ ) -1 A) ≤ Cγ -1 N γ,O m,r,p (A) . (3.2.41) (iv) For any τ ∈ [0, 2π] we have e -iτ K 0 Ae iτ K 0 ∈ A γ,O m,r and N γ,O m,r,p (e -iτ K 0 Ae iτ K 0 ) ≤ CN γ,O m,r,q (A) . (3.2.42)

Conjugation rules

Let ω • ∂ ϕ be the diagonal operator acting on sequences z ∈ s,r (see (3.2.13)) defined by

ω • ∂ ϕ z := diag l∈Z d ,k∈N (iω • l)z = (iω • lz [k] (l)) l∈Z d ,k∈N .
(3.2.43)

Consider an operator of the form

L := L(ϕ, ω) := ω • ∂ ϕ + iM (ϕ) , (3.2.44)
where M (ϕ) is some map

T d ϕ → M = M (ϕ) ∈ L(H s ; H s+m ),
for some m ∈ R. We shall study how the operator L in (3.2.44) conjugates under the map Φ S defined as

Φ S := (Φ τ S ) |τ =1 , Φ τ S := e iτ S = ∞ p=0 1 p! (iτ S) p , (3.2.45)
where S(ϕ) is some map T d ϕ → S = S(ϕ) ∈ L(H s ; H s+m ), for some m ∈ R. For the well-posedness of a map of the form (3.2.45) we refer to Lemma 3.6.6 in Appendix 3.6.3.

By using the Lie series expansions we have

L + = L + (ϕ) := Φ S • L • Φ -1 S = ω • ∂ ϕ + iM + (ϕ) , (3.2.46)
where M + (ϕ) = M + 1 (ϕ) + M + 2 (ϕ) with, for any q ∈ N,

iM + 1 (ϕ) := Φ S • iM • Φ -1 S = iM + q p=1 1 p! ad p iS (iM ) + 1 q! 1 0 (1 -τ ) q Φ τ S ad q+1 iS (iM )Φ -τ S dτ , (3.2.47) and iM + 2 (ϕ) := Φ S • ω • ∂ ϕ • Φ -1 S -ω • ∂ ϕ = -iω • ∂ ϕ S - q p=2 1 p! ad p-1 iS (iω • ∂ ϕ S) + 1 q! 1 0 (1 -τ ) q Φ τ S ad q iS (iω • ∂ ϕ S)Φ -τ S dτ

Linear operators and matrices.

According to the orthogonal splitting

L 2 (M n ) = k∈N E k ,
we identify a linear operator acting on L 2 (M n ) with its matrix representation A :=

A [k ] [k] k,k ∈N in L(h 0 ) (recall (3.2.9)) with blocks A [k ] [k] ∈ L(E k ; E k ). Notice that each block A [k ] [k] is a d k × d k : A [k ] [k] := A k ,j k,j j=1,...,d k , j =1,...,d k . (3.2.50)
The action of the operator A on functions u(x) as in (3.2.7) of the space variable in L 2 (M n ) is given by

(Au)(x) = k∈N (Az) [k] • Φ [k] (x) , z [k] ∈ C d k , (Az) [k] = j∈N A [j] [k] z [j] .
(3.2.51)

Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential

In this paper we also consider regular ϕ-dependent families of linear operators

T d ϕ → A = A(ϕ) = l∈Z d A(l)e il•ϕ (3.2.52)
where A(l) are linear operators in L(H s , H s ), for any l ∈ Z d . We also regard A as an operator acting on functions u(ϕ, x) of space-time as (Au)(ϕ, x) = (A(ϕ)u(ϕ, •))(x). More precisely, expanding u as in (3.2.12), we have

(Au)(ϕ, x) = l∈Z d ,k∈N (Az) [k] (l)e il•ϕ Φ [k] (x) , (Az) [k] (l) = p∈Z d ,k ∈N A [k ] [k] (l -p)z [k ] (p) .
(3.2.53) Relation (3.2.51) shows that, in order to define operators that conserve the H s regularity in space we need to assume some decay of

A [k ] [k] 2 L(L 2 ) with respect to |k -k |.
That the reason for the following definition first introduced in [BP11] for (i) and in [START_REF] Berti | An Abstract Nash Moser Theorem and Quasi-Periodic Solutions for NLW and NLS on Compact Lie Groups and Homogeneous Manifolds[END_REF] for (ii).

Definition 3.2.8. (s-decay norm).

(i) We define the s-decay norm of a matrix A ∈ L(H s ; H s ) as

|A| 2 s := h∈N h 2s sup |k-k |=h A [k ] [k] 2 L(L 2 ) (3.2.54)
where

• L(L 2 ) is the L 2 -operator norm in L(E k , E k ). (ii) Consider a map T d ϕ → A = A(ϕ) ∈ L(H s ; H s ).
We define its decay norm as

[[A]] 2 s := l∈Z d ,h∈N l, h 2s sup |k-k |=h A [k ] [k] (l) 2 L(L 2 ) . (3.2.55)
We denote by M s the space matrices with finite s-decay norm

[[•]] s . (iii) Consider a Lipschitz family O ω → A(ω) ∈ M s where O is a compact subset of R d , d ≥ 1.
For γ > 0 we define the Lipschitz decay norm as

[[A]] γ,O s := [[A]] sup,O s + γ[[A]] lip,O s = sup ω∈O [[A(ω)]] s + γ sup ω 1 ,ω 2 ∈O ω 1 =ω 2 [[A(ω 1 ) -A(ω 2 )]] s |ω 1 -ω 2 | . ( 3
M s = ∩ p+q≤s H p (T d , L dec (H q , H q )) ,
where L dec (H q , H q ) are bounded operator from H q to H q with finite | • | s -norm (see (3.2.54)).

Remark 3.2.10. Notice that, if the s-decay norm of a matrix A is finite, then

A [k ] [k] L(L 2 ) ≤ C(s)[[A]] s k -k -s .
We have the following fundamental lemma stating in particular that the s-decay norm is tame (see (3.2.58)). This tame property will be crucial in the KAM procedure. Lemma 3.2.11. For any s > (d + n)/2 the following holds :

(i) there is C = C(s) > 0 such that (recall (3.2.14),(3.2.15)) Az s ≤ C[[A]] s z s 0 + C[[A]] s 0 z s , (3.2.

57)

for any h ∈ s ;

(ii) there is C = C(s) > 0 such that

[[AB]] s ≤ C[[A]] s [[B]] s 0 + C[[A]] s 0 [[B]] s ; (3.2.58) (iii)
for N > 0 we define (recall (3.2.52)) the matrix Π N A as

(Π N A) [k ] [k] (l) :=            A [k ] [k] (l) , l ∈ Z d , k, k ∈ N , |l| ≤ N , |k -k | ≤ N , 0 , otherwise (3.2.59)
One has

[[(Id -Π N )A]] s ≤ CN -β [[A]] s+β , β ≥ 0 , (3.2.60)
for some C = C(s) > 0.

Similar bounds holds also replacing

• s , [[•]] s with the norms • γ,O s , [[•]] γ,O s respectively (see (3.2.20), (3.2.56)).
Proof. Items (i) and (ii) follow by lemmata 2.6, 2.7 in [START_REF] Berti | An Abstract Nash Moser Theorem and Quasi-Periodic Solutions for NLW and NLS on Compact Lie Groups and Homogeneous Manifolds[END_REF]. Item (iii) follows by the definition of the norm in (3.2.55).

Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential

We will also need a class of matrices that take into account a notion of regularization. Definition 3.2.12. Define the diagonal ϕ-independent operator D, acting on z ∈ s (see (3.2.14)), as

Dz := diag l∈Z d ,k∈N λ k z = λ k z [k] (l) l∈Z d ,k∈N
.

(3.2.61)

For β ∈ R we define the norm [[•]] β,s of a matrix A in (3.2.52) as [[A]] β,s := [[D β A]] s + [[AD β ]] s . (3.2.62)
We denote by M β,s the space of maps

T d ϕ → A = A(ϕ) ∈ L(L 2 ) with finite [[•]] β,s - norm. Consider a family O ω → A(ω) ∈ M β,s where O is a compact subset of R d , d ≥ 1.
For γ > 0 we define the Lipschitz norm as

[[A]] γ,O β,s := [[A]] sup,O β,s + γ[[A]] lip,O β,s = sup ω∈O [[A(ω)]] β,s + γ sup ω 1 ,ω 2 ∈O ω 1 =ω 2 [[A(ω 1 ) -A(ω 2 )]] β,s |ω 1 -ω 2 | . ( 3.2.63)
We denote by M γ,O β,s the space of families of matrices

A(ω) with finite [[•]] γ,O β,s -norm.
For properties of matrices in M γ,O β,s we refer to Appendix 3.6.2 and in particular Lemma 3.6.4 stating a tame property for the norm given by (3.2.63).

We end this section with the following definition : Definition 3.2.13. (Block-diagonal matrices). We say that A(ϕ) is block-diagonal if and only if

A [k ] [k] (ϕ) = 0 for any k = k and any ϕ ∈ T d .
We notice that operators commuting with K 0 have matrices that are block-diagonal : let

Z be such that [K 0 , Z] = 0 . (3.2.64) Since [H 0 , Z] [k ] [k] = (λ k -λ k )Z [k ] [k]
∀k, k , condition (3.2.64) implies that the matrix (Z

[k ] [k]
)) k,k ∈N representing the operator Z is block-diagonal according to Definition 3.2.13.

Link between pseudo-differential operators and matrices

To a linear operator R we associate its matrix representation still denoted R through the formula

R [k ] [k] = M n RΦ [k] Φ [k ] dx.
(3.2.65)

In the following we show that the decay norm [[•]] β,s (see Definitions 3.2.8 and 3.2.12) is well designed to capture the smoothing property.

Lemma 3.2.14.

Fix s > (d + n)/2 and β ≥ 0. Assume that R ∈ R ρ,s with ρ ≥ s + β + 1/2 and that R is symmetric then R ∈ M β,s . Moreover, there exists a constant C = C(s, ρ, β) such that [[R]] β,s ≤ C|R| ρ,s,s (3.2.66) If R ∈ R γ,O ρ,s then the bound (3.2.66) holds with the norms [[•]] β,s, , | • | ρ,s,s replaced by the norms [[•]] γ,O β,s , | • | γ,O ρ,s,s . Proof. We have for l ∈ Z d ||R [k ] [k] (l)|| L(L 2 ) = | D ρ+s R(l)Φ [k] , D -ρ-s Φ [k ] | ≤ ||D ρ+s R(l)Φ [k] || L 2 ||Φ [k ] || L 2 k -ρ-s ≤ R(l) L(H s ,H s+ρ ) Φ [k] H s k -ρ-s ≤ R(l) L(H s ,H s+ρ ) k s k -ρ-s ,
where we used that, for s ∈ R (recall (3.2.10)),

Φ k,j H s ∼ K s 0 Φ k,j L 2 = λ s k ∼ k s .
Similarly, since R is symmetric,

||R [k ] [k] (l)|| L(L 2 ) ≤ R(l) L(H s ,H s+ρ ) k s k -ρ-s , therefore we get ||R [k ] [k] (l)|| L(L 2 ) ≤ min k s k -ρ-s , k s k -ρ-s R(l) L(H s ,H s+ρ ) .

Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential

So, by definition, we get using that h, l ≤ l h ,

[[D β R]] 2 s = h∈N,l∈Z d h, l 2s sup |k-k |=h (D β R) [k ] [k] (l) 2 L(L 2 ) ≤ l∈Z d l 2s R(l) L(H s ,H s+ρ ) h∈N h 2s sup |k-k |=h k 2β min k s k -ρ-s , k s k -ρ-s ≤ 2 2ρ-2β |R| 2 ρ,s,s h∈N h 2s+2β-2ρ
where we used that if

|k -k | = h then max(|k|, |k |) ≥ h/2. A similar estimates holds true for [[RD β ]] s and thus for [[R]] β,s = [[D β R]] s + [[RD β ]] s .
Following a similar reasoning one gets the Lipschitz bounds.

Regularization procedure

Let us consider 0 < δ < 1, r > d/2 and the operator

F = F(ω) := ω • ∂ ϕ + i(∆ g + V (ϕ)) , V ∈ A δ,r . (3.3.1)
We also assume that the operator V is self-adjoint. Let us define the diophantine set

O 0 ⊆ [1/2, 3/2] d by O 0 := ω ∈ [1/2, 3/2] d : |ω • l| ≥ 4γ|l| -τ , ∀ l ∈ Z d , τ := d + 1 . (3.3.2)
The aim of this section is to prove the following result.

Theorem 3.3.1. (Regularization). Let ρ 0 ≥ 0, 0 < δ < 1 and r 0 > d/2. There is r * = r * (δ, ρ 0 , r 0 ) such that, for r > r * and S ≥ s 0 > n/2, there exist p = p(S, ρ 0 ) ≥ 1 and 0 < ε * = ε * (S, ρ 0 ) such that the following holds. If

γ -1 N δ,r,p (V ) ≤ ε * . (3.3.3)
then there is, for any ϕ ∈ T d , for any ω ∈ O 0 , a bounded and invertible map Φ ∈ L(H s , H s ) for any s ∈ [s 0 , S] such that

F + := Φ • F • Φ -1 := ω • ∂ ϕ + i(∆ g + Z + R) , (3.3.4) where Z ∈ A γ,O 0 δ is independent of ϕ, Z is Hermitian and [Z, K 0 ] = 0 , (3.3.5) 3.3. Regularization procedure R(ϕ) is a Hermitian ρ 0 -smoothing operator in R γ,O 0 ρ 0 ,r 0 . Furthermore Z = Z 1 + Z 2 with Z 1 ∈ A δ is independent of ω ∈ O 0 , and Z 2 ∈ A γ,O 0
2δ-1 . Moreover the following estimates holds : for any s ∈ [s 0 , S] there exit constants q = q(s, ρ 0 ) ≥ p and C = C(s, ρ 0 ) > 0 such that

N δ,s (Z 1 ) + N γ,O 0 2δ-1,s (Z 2 ) ≤ CN δ,r,q (V ) , (3.3.6) |R| γ,O 0 ρ 0 ,r 0 ,s ≤ CN δ,r,q (V ) , (3.3.7) sup ϕ∈T d Φ ±1 (ϕ) -Id L(H s ,H s-δ ) ≤ CN δ,r,q (V ) , (3.3.8) sup ϕ∈T d Φ ±1 (ϕ) L(H s ,H s ) ≤ 1 + CN δ,r,q (V ) . (3.3.9)
As explained in the introduction this Theorem will be demonstrated by an iterative procedure alternating an averaging step according to the periodic flow of K 0 (section 3.3.1) and a step of eliminating the time dependence of the averaged term (section 3.3.2). The iteration is detailed in section 3.3.3.

Averaging procedure

For A ∈ A m , m ∈ R, we denote for τ ∈ [0, 2π] A(τ ) := e -iτ K 0 Ae iτ K 0 (3.3.10)
and

A := 2π 0 A(τ )dτ , (3.3.11)
the average of A along the flow of K 0 . We notice that A belongs to A m , commutes with K 0 and that if A is Hermitian then

A is Hermitian. Let O ⊂ O 0 (see (3.3.2))
and consider the operator

G = ω • ∂ ϕ + iM (ϕ) M (ϕ) := ∆ g + W + A(ϕ) + R(ϕ) (3.3.12)
where W ∈ A γ,O δ , 0 < δ < 1, is independent of time and commutes with K 0 , A ∈ A γ,O δ ,r for some δ ≤ δ and R(ϕ) ∈ R γ,O ρ,r (see Def. 3.2.4). We also assume that M (ϕ) is Hermitian

∀ ϕ ∈ T d . Lemma 3.3.2. Let r > d/2, 0 < δ < 1, δ ≤ δ there exists S ∈ A γ,O
δ -1,r such that for any s > n/2 and ρ ≥ 0 there exists p = p(s, ρ) ≥ 1, an increasing function of s, and Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential

0 < ε 0 = ε 0 (s, ρ) such that if γ -1 N γ,O δ ,r,p (A) ≤ ε 0 , N γ,O δ,r,p (W ) ≤ 1 (3.3.13)
the symplectic change of variable Φ S = e iS(ϕ) belongs to L(H s , H s ) and we have

G + := Φ S • G • Φ -1 S = ω • ∂ ϕ + iM + (ϕ) (3.3.14) M + (ϕ) := ∆ g + W + A(ϕ) + A + (ϕ) + R + (ϕ) (3.3.15) where A(ϕ) is defined as in (3.3.11), A + ∈ A γ,O δ -1,r-1 and R + ∈ R γ,O ρ,r-1 . The operator M + (ϕ) is Hermitian ∀ ϕ ∈ T d . Moreover there exists C = C(s, ρ) such that N γ,O δ -1,r,s (S) ≤ CN γ,O δ ,r,p (A) (3.3.16) sup ϕ∈T d Φ τ S (ϕ) γ,O L(H s ,H s ) ≤ 1 + CN γ,O δ ,r,p (A) (3.3.17) sup ϕ∈T d Φ τ S (ϕ) -Id γ,O L(H s ,H s ) ≤ CN γ,O δ ,r,p (A) ∀τ ∈ [0, 1] , (3.3.18) N γ,O δ -1,r-1,s (A + ) ≤ CN γ,O δ ,r,p (A) (3.3.19) |R + | γ,O ρ,r-1,s ≤ C|R| γ,O ρ,r,s + CN γ,O σ,δ ,p (A) . (3.3.20)
Proof. The idea comes from [START_REF] Weinstein | Asymptotics of eigenvalue clusters for Laplacian plus a potential[END_REF], [START_REF] De Verdière | Sur le spectre des opérateurs elliptiques à bicaractéristiques toutes périodiques[END_REF] and was extensively used in [START_REF] Bambusi | Growth of Sobolev norms for abstract linear Schrödinger equations[END_REF]. It consists to average with respect to the flow of K 0 (see (3.3.11)) which is periodic since its spectrum is included in

N + λ (see (3.2.1)). Let us define Y = 1 2π 2π 0 τ (A -A )(τ )dτ . Then Y ∈ A γ,O δ ,r
and by integration by parts we verify that Y solves the homological equation

i[K 0 , Y ] = A -A .

Then we define

S = 1 4 (Y K -1 0 + K -1 0 Y ) (3.3.21)
and we note that S ∈ A γ,O δ -1,r is a pseudo-differential operator of order δ -1 ≤ 0. Moreover, by using Lemma 3.2.6, we deduce the estimate (3.3.16). By applying Lemma 3.6.6 we obtain estimates (3.3.17) and (3.3.18) (see (3.6.16), (3.6.17)). By an explicit computation we also get

i[K 2 0 , S] = A -A - 1 4 [A, K 0 ], K -1 0 . (3.3.22)
To study the conjugate of G in (3.3.12) under the map Φ S defined as in (3.2.45) with S in (3.3.21) we use the Lie expansions (3.2.47) and (3.2.48) for some q ∈ N large to be chosen later. Recalling the splitting (3.2.1)-(3.2.2) we have by (3.2.47)

Φ S • iM • Φ -1 S (3.3.22) = iK 2 0 + iQ 0 + iW + i A + i 4 [A, K 0 ], K -1 0 -i[Q 0 + W + A, iS] + q j=2 1 j! ad j iS (i∆ g + iW + iA) + 1 q! 1 0 (1 -τ ) q e iτ S ad q+1 iS (i∆ g + iW + iA)e -iτ S dτ + iΦ S • R • Φ -1 S .
Taking into account the time contribution given by (3.2.48) we obtain that the conjugate

Φ S • G • Φ -1 S has the form (3.3.14)-(3.3.15) where iA + = i 4 [A, K 0 ], K -1 0 -i[Q 0 + W + A, iS] + q j=2 1 j! ad j iS (i∆ g + iW + iA) - q p=1 1 p! ad p-1 iS (iω • ∂ ϕ S) (3.3.23) and iR + = 1 q! 1 0
(1 -τ ) q e iτ S ad q+1 iS (i∆ g + iW + iA)e -iτ S dτ

+ 1 q! 1 0 (1 -τ ) q Φ τ S ad q iS (iω • ∂ ϕ S)Φ -τ S dτ , + iΦ S • R • Φ -1 S (3.3.24)
We need to prove the bounds (3.3.19)-(3.3.20). We start by studying the remainder R + in (3.3.24). To simplify the notation we shall write a b to denote a ≤ Cb for some constant

C = C(s, ρ).
Using the smallness condition (3.3.13), we have that the third summand in (3.3.24) is a ρ-smoothing operator satisfying (3.3.20) by Lemma 3.6.8. By items (ii), (iii) of Lemma 3.2.6 we have (up to smoothing remainder and for some p depending on s and ρ)

N γ,O δ ,r,s ad iS (i∆ g + iW + iA) + N γ,O δ -1,r-1,s (ω • ∂ ϕ S) (3.3.13),(3.3.16) N γ,O δ ,r,p (A) . (3.3.25)
By iterating the estimate above and using the smallness condition (3.3.13) we deduce, for Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential 1 ≤ j ≤ q and for some p depending on s, ρ, q,

N γ,O jδ -2(j-1),r,s ad j iS (i∆ g + iW + iA) + N γ,O (j+1)δ -1-2j,r-1,s (ad j iS (ω • ∂ ϕ S)) N γ,O δ ,r,p (A) . (3.3.26)
The sequences jδ -2(j -1) and (j + 1)δ -1 -2j are decreasing since δ ≤ 1. Hence, by choosing q large enough, the integrands in (3.3.24) are ρ-smoothing operator (with arbitrary ρ) conjugated by the flow e iτ S . Therefore by Lemma 3.6.8 all the expressions in (3.3.24) are smoothing remainders satisfying (3.3.20) for some p depending on s and ρ.

Let us now consider the terms in (3.3.23). First of all we have

N γ,O δ -2,r,s ( [A, K 0 ], K -1 0 ) N γ,O δ ,r,N 1 ([A, K 0 ])N γ,O -1,N 1 (K -1 0 ) N γ,O 0 δ ,r,N (A)N γ,O 1,N (K 0 )N γ,O -1,N (K -1 0 ) N γ,O 0 δ ,r,N (A) ,
for some constant N 1 ≤ N ≤ p depending only on s, ρ. In the same way (recalling also (3.3.13)) we have

N γ,O δ+δ -2,r,s ([Q 0 + W + A, iS]) N γ,O δ -1,r,p (S) N 0,p (Q 0 ) + N γ,O δ,p (W ) + N γ,O δ -1,r,p (S)N γ,O δ ,r,p (A) (3.3.13),(3.3.16) N γ,O δ ,r,p (A) .
The other summands in (3.3.23) can be estimated by using (3.3.25) and (3.3.26). This proves the (3.3.19).

Time elimination

Let us consider the operator G + in (3.3.14)-(3.3.15) obtained after an average step (see Lemma 3.3.2). The aim of this section is to eliminate the time dependence (i.e. the dependence with respect to ϕ) in the term A(ϕ) in (3.3.15). First we introduce the pseudo-differential operator T = T (ϕ) defined as

T (ϕ) = 0 =l∈Z d e il•ϕ iω • l A(l) . (3.3.27)
We have the following Lemma.

A(ϕ) -ω • ∂ ϕ T = A(0) , (3.3.28)
and satisfies

N γ,O 0 δ ,r-(2τ +1),s (T ) ≤ CN γ,O 0 δ ,r,p (A) . (3.3.29)
Furthermore, setting Φ τ T := e iτ T (ϕ) , we have that for any s > d/2 there are constants C, p (depending only on s and ρ ) such that if (3.3.13) holds then

sup ϕ∈T d Φ τ T L(H s ,H s ) ≤ 1 + CN γ,O 0 δ ,r,p (A) (3.3.30) sup ϕ∈T d Φ τ T -Id L(H s ,H s-δ ) ≤ CN γ,O 0 δ ,r,p (A) ∀τ ∈ [0, 1] . (3.3.31)
Proof. The operator T is Hermitian and commutes with K 0 thanks to the properties of A . The fact that T solves (3.3.28) is obtained by an explicit computation. The bound (3.3.29) follows by item (iii) of Lemma 3.2.6. Finally applying Lemma 3.6.6 we obtain the estimates (3.3.30)-(3.3.31) (see (3.6.18) and (3.6.19)).

In the following lemma we study how the operator G + in (3.3.14)-(3.3.15) changes under the map Φ τ T defined by Lemma 3.3.3. We have to distinguish the cases δ strictly positive or δ less or equal zero. Lemma 3.3.4. Let δ ≤ 0 and r > 2τ + 2 + d/2. Let us define δ 1 := δ + δ -1 and

Φ T := Φ 1 T . Then the conjugated operator G 1 := Φ T • G + • Φ -1 T has the form G 1 = ω • ∂ ϕ + iM 1 (ϕ) (3.3.32) M 1 (ϕ) := ∆ g + W 1 + A 1 (ϕ) + R 1 (ϕ) (3.3.33)
where

W 1 = W + T d A(ϕ) dϕ , (3.3.34) is independent of ϕ ∈ T d , A 1 ∈ A γ,O δ 1 ,r-2τ -2 and R 1 ∈ R γ,O ρ,r-2τ -2 . The operator M 1 (ϕ) is Hermitian ∀ ϕ ∈ T d .
Moreover for any s > d/2 there exist p = p(s, ρ) and C = C(s, ρ) such that if (3.3.13) Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential holds then

N γ,O δ 1 ,r-2τ -2,s (A 1 ) ≤ CN γ,O δ ,r,p (A) (3.3.35) |R 1 | γ,O ρ,r-2τ -2,s ≤ C|R| γ,O ρ,r,s + N γ,O δ ,r,p (A) . (3.3.36) Proof. Notice that, since T in (3.3.27) commutes with K 0 , then Φ T • K 2 0 • Φ -1 T = K 2 0 .
:= iA + + q j=1 1 j! ad j iT iQ 0 + iW + i A(ϕ) + iA + - q j=2 1 j! ad j-1 iT (iω • ∂ ϕ T ) , (3.3.37) iR 1 := iΦ T • R + • Φ -1 T + 1 q! 1 0 (1 -τ ) q Φ τ T ad q+1 iT iQ 0 + iW + i A(ϕ) + iA + Φ -τ T dτ + 1 q! 1 0 (1 -τ ) q Φ τ T ad q iT (iω • ∂ ϕ T )Φ -τ T dτ , (3.3.38)
and where q ∈ N is a large constant to be chosen later. We now estimate the different terms in (3.3.37), (3.3.38).

By (3.2.38) we have, for some p = p (s, ρ),

N γ,O 0 2δ -1,r-2τ -2,s (ad iT (iω • ∂ ϕ T )) N γ,O 0 δ ,r-2τ -2,p (T )N γ,O 0 δ ,r-2τ -2,p (ω • ∂ ϕ T ) .
On the other hand we have by (3.2.39)

N γ,O 0 δ ,r-2τ -2,p (ω • ∂ ϕ T ) N γ,O 0 δ ,r-2τ -1,p (T ) ,
thus using (3.3.29) we deduce

N γ,O 0 2δ -1,r-2τ -2,s (ad iT (iω • ∂ ϕ T )) N γ,O 0 δ ,r,p (A) 2 (3.3.13) N γ,O 0 δ ,r,p (A) . (3.3.39)
Similarly we prove

N γ,O 0 δ -1,r-2τ -2,s (ad iT (Q 0 )) + N γ,O 0 δ+δ ,r-2τ -2,s (ad iT (W + A(ϕ) )) + N γ,O 0 2δ -2,r-2τ -2,s (ad iT (A + )) N γ,O 0 δ ,r,p (A) . (3.3.40)
Notice that, since 0 < δ < 1, the highest order pseudo-differential operator among the ones estimated in (3.3.39), (3.3.40) is the one of order δ + δ -1 < δ . By the estimates above, by choosing the constant q ∈ N large enough with respect to ρ and by reasoning Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential

(see (3.3.14)) G 0 := Φ S 0 • G 0 • Φ -1 S 0 = ω • ∂ ϕ + i∆ g + i V (ϕ) + i A 0 + i R 0 , A 0 ∈ A γ,O δ-1,r-1 , R 0 ∈ R γ,O ρ,r-1 , (3.3.44)
with ρ > 0 arbitrary to be chosen later and where V (ϕ) is defined as in (3.3.11). We apply Lemma 3.3.5 to the operator given by (3.3.44) with ρ 1 ρ 0 of Theorem 3.3.1 and r 1 > d/2 (to be chosen later) provided that ρ and r are sufficiently large (ρ > ρ 0 + δr 1 + 1 and r > max(r 1 + d/2, 2τ + 2 + d/2)). Hence we obtain a symplectic map Φ T 0 such that

G 1 := Φ T 0 • G 0 • Φ -1 T 0 = Φ T 0 • Φ S 0 • G 0 • Φ -1 S 0 • Φ -1 T 0 = ω • ∂ ϕ + i∆ g + iW 1 + iA 1 + iR 1 with W 1 := T d V (ϕ) dϕ, A 1 ∈ A γ,O 0 2δ-1,r 1 , R 1 ∈ R γ,O 0 ρ 0 ,
r 1 and estimates (3.3.41) (3.3.42) are satisfied for all s ∈ [s 0 , S] provided p ≥ p 2 (p 1 , S) depending on p 1 and S (and still increasing in S). We notice that W 1 is independent of ϕ and of the parameters ω ∈ O 0 . Now we want to iterate this procedure.

Let us first consider the case 6 0 < δ ≤ 1/2. Then 2δ -1 ≤ 0 and hence, form now on, we will apply iteratively Lemmata 3.3.2 and 3.3.4 (instead of Lemma 3.3.5).

We introduce the following parameters : for n ≥ 1 we set

δ n = (n + 1)δ -n , r n = r 1 -n(2τ + 2) , q n = q 0 • q n-1 (3.3.45)
where q 0 (•) = q 1 (•, S) is the composition of the two function s → p(s) given by Lemmata 3.3.2 and 3.3.4 and q 1 = p 2 • p 1 . We notice that q n is an increasing function of S.

Then applying Lemmata 3.3.2 and 3.3.4 iteratively, there exist symplectic changes of variables {Φ Sn } n and {Φ Tn } n such that, setting Φ n := Φ Tn • Φ Sn , we have

G n+1 := Φ n • G n • Φ -1 n = ω • ∂ ϕ + i∆ g + iW n+1 + A n+1 + R n+1 (3.3.46)
where W n is pseudo-differential operator independent of ϕ of order δ commuting with K 0 ;

A n is pseudo-differential operator of order δ n ; R n is ρ 0 -smoothing operator. Moreover, by estimates (3.3.19), (3.3.20) and (3.3.35), (3.3.36) we get

N γ,O 0 δ,s (W n ) + N γ,O 0 δn,rn,s (A n ) + |R n | γ,O 0 ρ 0 ,rn,s ≤ CN δ,r,qn (V ) for all s ∈ [s 0 , S] .
(3.3.47)

6. Actually Theorem 3.3.1 will be applied only in this case.

Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential

that c 0 is defined in (3.2.6)) |µ (0) k,j (ω)| ≤ c 0 2 |k| , ω ∈ O, k ∈ N, j = 1, • • • , d k , (3.4.3) (Z 0 ) [k] [k] lip,O L(L 2 ) ≤ 1 4 k -κ , k ∈ N . (3.4.4) (A2) the operator R 0 is in M γ,O
ρ,S (see Def. 3.2.12) and is Hermitian.

Let us define

:= γ -1 [[R 0 ]] γ,O 0 ρ,s 0 +b . (3.4.5)
We shall prove the following.

Φ ∞ = Φ ∞ (ω, ϕ) : H s → H s such that for all ϕ ∈ T d , for all ω ∈ O , L ∞ := Φ ∞ • L 0 • Φ -1 ∞ := ω • ∂ ϕ + i(∆ g + Z ∞ ) . (3.4.8)
Moreover we have

[[Φ ±1 ∞ (ϕ) -Id]] γ,O∞ ρ,s ≤ Cγ -1 [[R 0 ]] γ,O ρ,s+b , ∀ ω ∈ O , (3.4.9) [[ Z∞ ]] γ,O ρ,s ≤ C[[R 0 ]] γ,O ρ,s+b .
(3.4.10)

The KAM step

The proof of Theorem 3.4.1 is based on an iterative scheme. In this section we show how to perform one step of the iteration. We consider an operator

L := ω • ∂ ϕ + i(∆ g + Z + R) , (3.4.11)
where Z = Z 0 + Z 2 is Hermitian with Z 0 satisfying (A1) and Z 2 ∈ M γ,O ρ,s for all s ∈ [s 0 , S] and for some O ⊆ O 0 (see (3.3.2)). The remainder R satisfies (A2), i.e. belongs to M γ,O ρ,s for all s ∈ [s 0 , S] and is Hermitian.

Control of the small divisors

Let us denote by µ k,j , k ∈ N and j = 1, . . . , d k (see (3.2.3)), the eigenvalues of the block (∆ g + Z)

[k] [k]
. First of all we prove the following. Lemma 3.4.2. One has

sup k∈N k κ |µ [k] | lip,O ≤ 1 4 + [[Z 2 ]] lip,O κ,s 0 .
(3.4.12)

Proof. By Corollary A.7 in [START_REF] Feola | Reducibility of Schrödinger equation on the sphere[END_REF] the Lipschitz variation of the eigenvalues of an Hermitian matrix is controlled by the Lipschitz variation of the matrix. Then, in view of hypothesis (A1), we get

|µ [k] | lip,O 0 ≤ (Z 0 ) [k] [k] lip,O L(L 2 ) + (Z 2 ) [k] [k] lip,O 0 L(L 2 ) ≤ k -κ ( 1 4 + [[Z 2 ]] lip,O 0 κ,s 0 )
and the (3.4.12) follows.

We define the set O + ⊆ O of parameters ω for which we have a good control of the small divisors. We set, for N ≥ 1, for some constant C > 0 depending only on d.

O + ≡ O + (γ, N ) := ω ∈ O : |ω • l + µ k,j -µ k ,j | ≥ 2γ N τ k, k 2n+2 , |l| ≤ N, k, k ∈ N , j = 1, . . . , d k , j = 1, . . . , d k , (l, k, k ) = (0, k, k) .
Proof. We write

O \ O + = l∈Z d ,|l|≤N k,k ∈N ( ,k,k ) =(0,k,k) j=1,...,d k j =1,...,d k R j,j l,k,k where R j,j l,k,k := ω ∈ O : |ω • l + µ k,j -µ k ,j | ≤ 2γ N τ k, k 2n+2 .
Notice that when l = 0 and k = k then R j,j l,k,k = ∅ for all j, j . Indeed in such case we get using (3.4.3), (3.2.6) and (3.4.14)

|µ k,j -µ k ,j | ≥ c 0 2 (k + k ) -2[[Z 2 ]] ∞,O κ,s 0 +b ≥ c 0 2 - γ 4 ≥ 2γ.
Let us now consider the case l = 0. We give the estimate of the measure of a single bad set R j,j l,k,k . Let us consider the Lipschitz function

f (ω) = ω • l + µ k,j (ω) -µ k ,j (ω) = ω • l + g(ω) .
Using condition (3.4.14) we have that Lemma 3.4.2 implies that (recall that l = 0)

|g| lip,O ≤ 1 2 .
Then Lemma 5.2 in [START_REF] Feola | Reducibility of Schrödinger equation on the sphere[END_REF] implies that meas(R j,j l,k,k ) ≤ Cγ N τ k,k 2n+2 for some constant C > 0 depending only on d. Finally by (3.2.3) we have that

d k d k ≤ k, k 2(n-1) . Hence meas O \ O + ≤ C l∈Z d ,0<|l|≤N k,k ∈N j=1,...,d k j =1,...,d k R j,j l,k,k ≤ C l∈Z d ,0<|l|≤N k,k ∈N 2 γ N τ k, k 4 ≤ CN -1 γ 3.4. KAM reducibility since τ = d + 1.

Resolution of the Homological equation

In this section we solve the following homological equation

-iω • ∂ ϕ S + iS, ∆ g + Z + R = DiagR + Q (3.4.16)
where Q is some remainder to be determined and DiagR = (DiagR)

[k ]

[k] (l)

l∈Z d ,k,k ∈N , (DiagR) [k ] [k] (l) := 0 for l = 0 , k, k ∈ N or l = 0 , k = k , (DiagR) [k] [k] (0) := A [k] [k] (0) , otherwise .
(3.4.17

[[S]] γ,O + s ≤ s N 2τ +1 γ [[R]] γ,O ρ,s , [[D ±ρ SD ∓ρ ]] γ,O + s ≤ s N 2τ +ρ+1 γ [[R]] γ,O ρ,s , s ∈ [s 0 , S] , (3.4.18) [[Q]] γ,O + ρ,s ≤ s [[R]] γ,O ρ,s+b N -b , [[Q]] γ,O + ρ,s+b ≤ s [[R]] γ,O ρ,s+b , s ∈ [s 0 , S -b] . (3.4.19)
Proof. For N > 0 we define (recall (3.2.52)) the matrix Π N R as

(Π N R) [k ] [k] (l) :=            R [k ] [k] (l) , l ∈ Z d , k, k ∈ N , |l| ≤ N , |k -k | ≤ N ,
G(l, k, k , ω)A := -i ω • l + ∆ g + Z [k] [k] A + iA ∆ g + Z [k ] [k ]
.

(3.4.23)

Now, since ∆ + Z [k] [k]
is Hermitian, there is a orthogonal

d k × d k -matrix U [k] such that U T [k] ∆ g + Z [k] [k] U [k] = D [k] := diag j=1,...,d k µ k,j ,
where µ k,j are the eigenvalues of the k-th block. By setting

S [k ] [k] (l) := U T [k] S [k ] [k] (l)U [k ] , R [k ] [k] (l) := U T [k] R [k ] [k] (l)U [k ] equation (3.4.22) reads -i ω • l + D [k] S [k ] [k] (l) + i S [k ] [k] (l)D [k ] + (Π N R) [k ] [k] (l) = 0 . (3.4.24)
For ω ∈ O + (see (3.4.13)) the solution of (3.4.24) is given by (recalling the notation (3.2.50))

S k ,j k,j (l) :=            -i R k ,j k,j (l) ω • l + µ k,j -µ k ,j, , |l| ≤ N , |k -k | ≤ N , (l, k, k ) = (0, k, k) , 0 , otherwise .
(3.4.25)

Since R is Hermitian it is easy to check that also S is Hermitian. Using the bound on the small divisors in (3.4.13) we have that

| S k ,j k,j (l)| ≤ γ -1 | R k ,j k,j (l)|N τ k, k 2n+2 . ( 3
S [k ] [k] (l) L(L 2 ) = S [k ] [k] (l) L(L 2 ) ≤ d k d k S [k ] [k] (l) ∞ (3.4.26),(3.2.3) ≤ γ -1 R [k ] [k] (l) N τ k, k 3n+1 .
(3.4.27)

KAM reducibility

We now estimates the decay norm of the matrix S. We have

[[S]] 2 s (3.4.27) ≤ γ -2 N 2τ l,h l, h 2s sup |k-k |=h R [k ] [k] (l) 2 k, k 6n+2 ≤ γ -2 N 2τ l,h l, h 2s sup |k-k |=h k≥k (D ρ R) [k ] [k] (l) 2 k (6n+2-2ρ) + γ -2 N 2τ l,h l, h 2s sup |k-k |=h k<k (RD ρ ) [k ] [k] (l) 2 k (6n+2-2ρ) ≤ s γ -2 N 2τ [[R]] 2 ρ,s , (3.4.28)
provided that ρ ≥ 3n + 1 which is true thanks to the choices in (3.4.2). Hence the bound (3.6.5) in Lemma 3.6.1 implies

[[D ±ρ SD ∓ρ ]] s ≤ s γ -1 N τ +ρ [[R]] ρ,s . (3.4.29)
To obtain (3.4.18), it remains to estimate the Lipschitz variation of the matrix S. We reason as in the proof of item (iii) of Lemma 3.2.6. To simplify the notation, for any

l ∈ Z d , k, k ∈ N, j = 1, . . . , d k and j = 1, . . . , d k , we set d(ω) := i(ω • l + µ k,j (ω) -µ k ,j (ω)) , ∀ω ∈ O + .
(3.4.30) By (3.4.25) we have that, for any ω 1 , ω 2 ∈ O + S k ,j k,j (ω 1 ; l) -S k ,j k,j (ω 2 ; l) = R k j k,j (ω 1 ; l) -R k j k,j (ω 2 ; l) d(ω 1 )

+ d(ω 1 ) -d(ω 2 ) d(ω 1 )d(ω 2 ) R k ,j k,j (ω 2 ; l) .
Using the (3.4.12), (3.4.4) we deduce

|d(ω 1 ) -d(ω 2 )| |ω 1 -ω 2 | |l| , ∀ω 1 , ω 2 ∈ O + , ω 1 = ω 2 .
Therefore, recalling (3.4.13), (3.2.16) and reasoning as in (3.4.26), (3.4.27), we get

S [k ] [k] (l) lip,O + L(L 2 ) γ -1 N τ k, k 3n+1 R [k ] [k] (l) lip,O + γ -2 N 2τ +1 k, k 5n+3 R [k ] [k] (l) sup,O .
Finally, reasoning as in (3.4.28) and using (3.2.55), we deduce

[[S]] lip,O + s ≤ s γ -1 N τ [[R]] lip,O ρ,s + γ -2 N 2τ +1 [[R]] sup,O ρ,s , (3.4.31) [[R + ]] γ,O + ρ,s+b ≤ s [[R]] γ,O ρ,s+b + γ -1 N 2τ +ρ+1 [[R]] γ,O ρ,s 0 [[R]]
L + := Φ • L • Φ -1 = ω • ∂ ϕ + i(∆ g + Z) + iR + i[iS, ∆ g + Z] -iω • ∂ ϕ S + i p≥1 i p p! ad p S (R) + i p≥2 i p-1 p! ad S [iS, ∆ g + Z] -ω • ∂ ϕ S .
Hence, equations (3.4.16), (3.4.35) lead to the following formula :

R + = Q + R + with Q := (1 -Π N )R satisfying (3.4.19) and R + := p≥2 i p-1 p! ad p-1 S DiagR + Q -R + p≥1 1 p! ad p S R . (3.4.40)
Thus, in order to prove (3.4.38) we need to estimate R + . Consider (for instance) the composition operator SR. In order to control the [[•]] γ,O + ρ,s -norm we shall bound the decay norm of D ρ SR. The estimates for for SRD ρ is the same. We have that

[[D ρ SR]] γ,O + s = [[D ρ SD -ρ D ρ R]] γ,O + s (3.2.58) ≤ s [[D ρ SD -ρ ]] γ,O + s [[D ρ R]] γ,O s 0 + [[D ρ SD -ρ ]] γ,O + s 0 [[D ρ R]] γ,O s (3.4.18) ≤ s γ -1 N 2τ +ρ+1 [[R]] γ,O ρ,s [[R]]

Iteration and Convergence

In this section we introduce a new constant a := b -2 = 6d + 15n + 18 .

(3.4.42)

For N 0 ≥ 1 we define the sequence (N ν ) ν≥0 by

N ν := N χ ν 0 , ν ≥ 0
Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential Furthermore using (3.4.33), (3.4.49) at rank ν, N ν-1 = N 2/3 ν and 2τ + 1 -2 3 a ≤ -1, we obtain the estimate (3.4.46) at rank ν + 1. This proves the (S1) ν+1 .

We finally set

L ν+1 := Φ ν+1 • L ν • Φ -1 ν+1 = ω • ∂ ϕ + i(∆ g + Z ν+1 + R ν+1 ) (3.4.53)
where the remainder R ν+1 is given by Lemma 3.4.7. We have

[[R ν+1 ]] γ,O ν+1 ρ,s (3.4.38) ≤ s N -b ν [[R ν ]] γ,Oν ρ,s+b + γ -1 N 2τ +ρ+1 ν [[R ν ]] γ,Oν ρ,s 0 [[R ν ]] γ,Oν ρ,s (3.4.49) ≤ s [[R 0 ]] γ,O 0 ρ,s+b N -b+1 ν + γ -1 [[R 0 ]] γ,O 0 ρ,s 0 N 2τ +ρ+1-4 3 a ν ≤ N -a ν [[R 0 ]] γ,O 0 ρ,s+b (3.4.54)
for N 0 large enough where we used that γ

-1 [[R 0 ]] γ,O 0 ρ,s 0 ≤ 1 (thanks to (3.4.6)) and b ≥ a + 2 , 2τ + ρ + 1 - 1 3 a ≤ -1.
The latter condition is implied by the choice of a in (3.4.42) recalling the (3.4.2). The (3.4.54) is the first estimate in (3.4.49) at step ν + 1. We now give the estimate in "high" norm. We have

[[R ν+1 ]] γ,O ν+1 ρ,s+b (3.4.39) ≤ s [[R ν ]] γ,Oν ρ,s+b + γ -1 N 2τ +ρ+1 ν [[R ν ]] γ,Oν ρ,s 0 [[R ν ]] γ,Oν ρ,s+b (3.4.49) ≤ s [[R 0 ]] γ,O 0 ρ,s+b N ν-1 1 + γ -1 [[R 0 ]] γ,O 0 ρ,s 0 +b N 2τ +ρ+1 ν N -a-1 ν-1 ≤ s N ν [[R 0 ]] γ,O 0 ρ,s+b (3.4.55)
for N 0 large enough depending on s and thanks to fact that 3τ + 3 2 ρ + 1 2a ≤ 0. This is the (S3) ν+1 . Now we have to verify the initial step : ν = 1. (S2) 1 and (S4) 1 are proved exactly in the way as in the inductive step. Now to proceed we have to construct Φ 1 but now (3.4.52) becomes

C(s)γ -1 N 2τ +1 0 [[R 0 ]] γ,O 0 ρ,s 0 (3.4.5) ≤ C(s) N 2τ +1 0 ≤ 1 2 (3.4.56)
which is less than 1 2 for and N 0 satisfying (3.4.43). Furthermore using (3.4.33) we obtain

[[Ψ 1 ]] γ,O 1 s ≤ C(s)γ -1 N 2τ +1 0 [[R 0 ]] γ,O ρ,s ≤ γ -1 N 2τ +2 0 [[R 0 ]] γ,O ρ,s
for N 0 large enough. This proves the (S1) ν+1 .

Then we set

L 1 := Φ 1 • L 0 • Φ -1 1 = ω • ∂ ϕ + i(∆ g + Z 1 + R 1 ) (3.4.57)
where the remainder R 1 is given by Lemma 3.4.7. We have

[[R 1 ]] γ,O 1 ρ,s (3.4.38) ≤ s N -b 0 [[R 0 ]] γ,O 0 ρ,s+b + γ -1 N 2τ +ρ+1 0 [[R 0 ]] γ,O 0 ρ,s 0 [[R 0 ]] γ,O 0 ρ,s (3.4.49) ≤ s [[R 0 ]] γ,O 0 ρ,s+b N -b+1 0 + N 2τ +ρ+1 0 ≤ N -a 0 [[R 0 ]] γ,O 0 ρ,s+b
Φ ν+1 := Φ 1 • Φ 2 • • • • Φ ν+1 = Φ ν Φ ν+1 = Φ ν (Id + Ψ ν+1 ) . (3.4.60)
We want to prove that ( Φ ν ) ν≥1 converges in M γ,O∞ s . Let us define

δ (ν) s := [[ Φ ν ]] γ,O∞ s . (3.4.61)
We have

δ (ν+1) s 0 (3.2.58) ≤ δ (ν) s 0 (1 + C[[Ψ ν+1 ]] γ,O∞ s 0 ) (3.4.46),(3.4.5) ≤ δ (ν) s 0 (1 + C N -1 ν ) . (3.4.62)
By iterating the (3.4.62) we get, for any ν,

[[ Φ ν ]] γ,O∞ s 0 ≤ (1 + [[Ψ 1 ]] γ,O∞ s 0 )Π j≥1 (1 + C N -1 ν ) ≤ 2 (3.4.63)
where we used the (3.4.46) to estimate [[Ψ 1 ]] γ,O∞ s 0 and we take N 0 large enough.

Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential

The high norm of Φ ν+1 is estimated by

δ (ν+1) s (3.2.58) ≤ δ (ν) s (1 + C(s)[[Ψ ν+1 ]] γ,O∞ s 0 ) + C(s)[[Ψ ν+1 ]] γ,O∞ s [[ Φ ν ]] γ,O∞ s 0 (3.4.46),(3.4.63) ≤ δ (ν) s (1 + C(s) N -1 ν ) + ε ν (3.4.64)
where

ε ν := C(s)γ -1 [[R 0 ]] γ,O 0 ρ,s+b N -1 ν .
By iterating (3.4.64), using Π j≥0 (1 + C(s) N -1 ν ) ≤ 2 for N 0 large enough, we obtain

[[ Φ ν ]] γ,O∞ s ≤ [[ Φ 1 ]] γ,O∞ s + 2 j≥1 ε j ≤ 1 + C(s)γ -1 [[R 0 ]] γ,O 0 ρ,s+b . (3.4.65)
Then we have

[[ Φ ν+1 -Φ ν ]] γ,O∞ s = [[ Φ ν Ψ ν+1 ]] γ,O∞ s (3.2.58) ≤ s [[ Φ ν ]] γ,O∞ s [[Ψ ν+1 ]] γ,O∞ s 0 + [[ Φ ν ]] γ,O∞ s 0 [[Ψ ν+1 ]]
≤ s (1 + γ -1 [[R 0 ]] γ,O 0 ρ,s+b ) N -1 ν + γ -1 [[R 0 ]] γ,O 0 ρ,s+b N -1 ν ≤ s γ -1 [[R 0 ]] γ,O 0 ρ,s+b N -1 ν . ( 3
Z ∞ = Z 0 + Z ∞,2 := Z 0 + lim ν→∞ Z ν,2 , (3.4.67)
The (3.4.10) follows again by (3.4.47). We also notice that (3.4.49) implies that R ν → 0 in M γ,O∞ ρ,s . Now by applying iteratively the (3.4.45) we have that

L ν = Φ ν • L 0 • Φ -1
ν . Hence, passing to the limit, we get L ν → ν→∞ L ∞ of the form (3.4.8) with Z ∞ given by (3.4.67).

Proof of Theorem 3.1.1

In this short section we merge the two previous sections to prove the reducibility of the Schrödinger equation (3.1.1) : Theorem 3.1.1. and thus the smallness condition (3.4.6) is satisfied provided that ε is small enough. We now prove that Z 0 satisfies assumption (A1) with κ := 2δ -1. First we note that, since δ ≤ 1/2 then κ ≤ 0. Moreover, by Theorem 3.3.1, we have that

Z 0 := Z = Z 1 + Z 2 with Z 1 ∈ A δ independent of ω ∈ O 0 , and Z 2 ∈ A γ,O 0 2δ-1 . Estimate (3.3.6) implies that for all s ∈ [s 0 , S] N δ,s (Z) ≤ CN δ,r,p (εW ) ≤ CN δ,r,p (W )ε 1-α .
Since S ≥ p(δ, 0) we deduce by (3.2.21) that

Z L(L 2 ,H -δ ) ≤ CN δ,r,p (W )ε 1-α ≤ c 0 2
for ε small enough which in turn implies that

|µ (0) k,j (ω)| ≤ c 0 2 |k| δ
and thus (3.4.3) holds true. Furthermore since Z 1 does not depend on ω, we have

(Z 0 ) [k] [k] lip,O L(L 2 ) = (Z 2 ) [k] [k] lip,O L(L 2 )
and thus (3.3.6) implies also (3.4.4) for ε small enough. Hence all the hypothesis of Theorem 3.4.1 are satisfied for L 0 = F + and this theorem provides a set of frequencies O such that, for ω ∈ O , there is a map Φ ∞ satisfying, (see estimates (3.4.9), (3.5.4))

[[Φ ±1 ∞ (ϕ) -Id]] γ,O∞ ρ,s S N δ,r,p (W )ε 1-α ∀ ω ∈ O , ( 3
ϕ∈T d Ψ ±1 (ϕ) -Id L(H s ,H s-δ ) ≤ γ -1 C s ε , sup ϕ∈T d Ψ ±1 (ϕ) L(H s ,H s ) ≤ 1 + γ -1 C s ε , (3.5.7)
for some C s > 0. This concludes the proof.

Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential

Thus

N m,r-α,p ((ω • ∂ ϕ ) -1 A) 2 (3.2.31) ≤ 0 =l∈Z d 1 |ω • l| 2 l 2(r-α) N 2 m,p (A(l)) (3.2.40) ≤ 1 γ 2 l∈Z d \{0} l 2(r-α) |l| 2α N 2 m,p (A(l)) ≤ C γ 2 0 =l∈Z d l 2r N 2 m,p (A(l)) = C γ 2 N m,
B(ω 1 ) -B(ω 2 ) = 0 =l∈Z d 1 iω 1 • l e il•ϕ A(ω 1 ; l) -A(ω 2 ; l) + 0 =l∈Z d (ω 1 -ω 2 ) • l i(ω 1 • l)(ω 2 • l) e il•ϕ A(ω 2 ; l).
Moreover, by using (3.2.40) and that O is compact, we have

(ω 1 -ω 2 ) • l i(ω 1 • l)(ω 2 • l) ≤ C 1 γ 2 |l| 2α+1 |ω 1 -ω 2 | .
Therefore reasoning as in (3.6.2) we get

N m,r-(2α+1),p (B(ω 1 ) -B(ω 2 )) |ω 1 -ω 2 | 1 γ N lip,O m,r,p (A) + 1 γ 2 N sup,O m,r,p (A) .
(3.6.3) Combining (3.6.2), (3.6.3) and recalling (3.2.17), (3.6.1) we obtained

N γ,O m,r-(2α+1),p (B) 1 γ N sup,O m,r,p (A) + γ N lip,O m,r,p (A) + 1 γ 2 N sup,O m,r,p (A) 1 γ N sup,O m,r,p (A) + γN lip,O m,r,p (A)
which is bound (3.2.41).

Properties of the s-decay norm

In this appendix s 0 is some fixed number satisfying s 0 > (d + n)/2. Lemma 3.6.1. Let α > 0. Then (recall (3.2.61), (3.2.55))

[[D ±α AD ∓α ]] γ,O s ≤ s [[A]] γ,O s+α , (3.6.4) [[D ±α (Π N A)D ∓α ]] γ,O s ≤ s N α [[A]] γ,O s . ( 3
A(ϕ) L(H s ,H s ) ≤ s |A(ϕ)| s ≤ s [[A]] s+s 0 , ∀ ϕ ∈ T d .
Proof. See Lemma 2.4 in [START_REF] Baldi | KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation[END_REF].

Lemma 3.6.3. Assume that

C(s)[[A]] γ,O s 0 ≤ 1/2 (3.6.6)
for some large C(s) > 0 depending on s ≥ s 0 . Then the map Φ := Id + Ψ defined as

Φ := e iA := p≥0 1 p! (iA) p , (3.6.7) satisfies [[Ψ]] γ,O s ≤ s [[A]] γ,O s .
(3.6.8)

Proof. For any n ≥ 1, using (3.2.58), we have, for some C(s) > 0,

[[A n ]] s 0 ≤ [C(s 0 )] n-1 [[A]] n s 0 , [[A n ]] s ≤ n[C(s)[[A]] s 0 ] n-1 C(s)[[A]] s , ∀ s ≥ s 0 .
The same holds also for the norm

[[•]] γ,O s . Hence [[Ψ]] γ,O s ≤ [[A]] γ,O s p≥1 C(s) p p! ([[A]] γ,O s 0 ) p-1 ,
for some (large) C(s) > 0. By the smallness condition (3.6.6) one deduces the bounds (3.6.8).

Lemma 3.6.4. Let α, β ∈ R. Then

[[AM ]] γ,O α+β,s ≤ s [[A]] γ,O α,s+|β| [[M ]] γ,O β,s 0 +|α| + [[A]] γ,O α,s 0 +|β| [[M ]] γ,O β,s+|α| , (3.6.9) [[(Id -Π N )M ]] γ,O β,s ≤ s N -s [[M ]] γ,O β,s+s , s ≥ 0 . (3.6.10) (ii) the map Φ τ 2 satisfies sup ϕ∈T d Φ τ 2 (ϕ) L(H s ,H s ) ≤ (1 + CN γ,O δ,r,p (S 2 )) , (3.6.18) sup ϕ∈T d (Φ τ 2 (ϕ) -Id) L(H s ,H s-δ ) ≤ CN γ,O δ,r,p (S 2 ) , (3.6.19) sup ϕ∈T d (∂ k ϕ Φ τ 2 )(ϕ) L(H s ,H s-kδ ) ≤ CN γ,O δ,r,p (S 2 ) , 1 ≤ k ≤ r , (3.6.20)
for any τ ∈ [0, 1] and any ω ∈ O. Moreover the following bounds on the Lipschitz norm hold true :

sup ϕ∈T d Φ τ 2 (ϕ) γ,O L(H s ,H s-1 ) ≤ (1 + CN γ,O δ,r,p (S 2 )) , (3.6.21) sup ϕ∈T d (Φ τ 2 (ϕ) -Id) γ,O L(H s ,H s-δ-1 ) ≤ CN γ,O δ,r,p (S 2 ) , (3.6.22) sup ϕ∈T d (∂ k ϕ Φ τ 2 )(ϕ) γ,O L(H s ,H s-kδ-1 ) ≤ CN γ,O δ,r,p (S 2 ) , 1 ≤ k ≤ r , (3.6.23) for any τ ∈ [0, 1].
Proof. We shall prove the result for the map Φ τ 2 . The estimates on Φ τ 1 can be obtained in the same way. Notice that the operator Φ τ 2 solves the problem

   ∂ τ Φ τ 2 (ϕ) = iS 2 (ϕ)Φ τ 2 (ϕ) Φ 0 2 (ϕ) = Id .
(3.6.24)

The existence of the flow Φ τ 2 in L(H s , H s ) can be obtained following the line of chapter 5 in [START_REF] Taylor | Pseudifferential Operators and Nonlinear PDEs, sous la dir[END_REF]. Then using (3.6.24) and the assumption (3.6.13) one can check that

∂ τ Φ τ 2 h 2 H s = 0 ⇒ Φ τ 2 h H s ≤ h H s ,
for any τ ∈ [0, 1], h ∈ H s and ϕ ∈ T d . This is the (3.6.18). Let us now define

Γ τ (ϕ) := Φ τ 2 (ϕ) -Id .
It solve the problem

∂ τ Γ τ (ϕ) = iS 2 (ϕ)Γ τ (ϕ) + iS 2 (ϕ) , Γ 0 (ϕ) = 0 . By Duhamel formula have Γ τ (ϕ)h = τ 0 Φ τ 2 Φ -σ 2 (ϕ)iS 2 (ϕ)hdσ .

4

Birkhoff normal form for abcd Boussinesq system on the circle

Introduction

We consider the abcd-Boussinesq system

   (1 -b∂ xx )∂ t η + ∂ x (a∂ xx u + u + uη) = 0 (t, x) ∈ R × T (1 -d∂ xx )∂ t u + ∂ x (c∂ xx η + η + 1 2 u 2 ) = 0 (4.1.1)
Where η, u are real functions with zero momentum T η(t, x) dx = T u(t, x) dx = 0.

Here the independent variable x corresponds to distance along the channel and t is proportional to elapsed time. The quantity η = η(x, t) corresponds to the depth of the water at the point x and time t. The variable u(x, t) is proportional to the horizontal velocity at the height θh, where θ is a fixed constant in the interval [0, 1] and h is the undisturbed water depth.

We are interested in the long time behavior of small amplitude solution of (4.1.1). In this paper, we would like to prove a Birkhoff normal form result for (4.1.1). The idea comes from Theorem 2.13 of [START_REF] Bambusi | Birkhoff normal form for partial differential equations with tame modulus[END_REF] (see also [START_REF] Bambusi | Birkhoff normal form for some nonlinear PDEs[END_REF], [START_REF] Bambusi | Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on Zoll manifolds[END_REF], [START_REF] Grébert | Birkhoff normal form and Hamiltonian PDEs[END_REF], [START_REF] Bambusi | A Birkhoff normal form theorem for some semilinear PDEs[END_REF], [START_REF] Delort | A quasi-linear Birkhoff normal forms method. Application to the quasi-linear Klein-Gordon equation on S 1[END_REF]), where the authors prove a Birkhoff normal form result for Hamiltonian with nonlinearity satisfying a tame modulus condition and nonresonant frequencies. In [START_REF] Bambusi | Birkhoff normal form for partial differential equations with tame modulus[END_REF], the authors applied that result to many partial differential equations : NLW equations, NLS equations in one dimension and higher dimension, coupled NLS in one dimension. In this paper, we state a similar result for Boussinesq systerm.

As in general, the proof of a Birkhoff normal form result is obtained by constructing a sequence of canonical transformations to eliminate the non-normalized part of the nonli-nearity step by step. This procedure relates to solving homological equations in each step. The idea in [START_REF] Bambusi | Birkhoff normal form for partial differential equations with tame modulus[END_REF] is to use a so-called tame inequality

uv s ≤ C( u s v 1 + v s u 1 ) (4.1.2)
and that if u ∈ H s depends only on high modes u = |k|≥N u k e i2πkx then

u 1 ≤ u s N s-1 . (4.1.3)
This term is small when N large enough. Then in the Birkhoff procedure, the nonlinearity, which satisfies the tame properties and has many high modes in expansion, is small and controllable. The study of frequency plays an important role in order to eliminate the remainder part of non-integrable resonant terms. However, contrary [START_REF] Bambusi | Birkhoff normal form for partial differential equations with tame modulus[END_REF], where external potentials were used to verify the nonresonant condition, we obtain the nonresonant condition by studying frequencies as functions of parameters a, b, c, d.

Our main contribution in this work is to verify that the nonlinearity in Boussinesq system satisfies a tame property and that the frequencies satisfy a nonresonant condition(section 4) then we prove our Birkhoff normal form result(section 5). In section 2, we state our main theorem and explain the scheme of our proof. In section 3, we recall notations of tame property and nonresonant condition, which was introduced in [START_REF] Bambusi | Birkhoff normal form for partial differential equations with tame modulus[END_REF]. We give the proof of local well-posedness in Appendix B.

The system (4.1.1) was originally derived by Bona, Chen and Saut [START_REF] Bona | Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media. I : Derivation and Linear Theory[END_REF], [START_REF] Lioyd Bona | Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media : II. The nonlinear theory[END_REF] in the vein of the Boussinesq original derivation [START_REF] Boussinesq | Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond[END_REF]. The equation is derived to study the two dimensional, incompressible and irrotational water wave in the shallow water regime. The abcd Boussinesq equation and its extensions have been studied extensively in the literature (see [START_REF] Alvarez-Samaniego | Large time existence for 3 D water-waves and asymptotics[END_REF], [START_REF] Bona | Long wave approximations for water waves[END_REF], [START_REF] Bona | Asymptotic models for internal waves[END_REF], [START_REF] Linares | Well-posedness of strongly dispersive two-dimensional surface wave Boussinesq sys-tems[END_REF], [START_REF] Saut | The Cauchy problem on large time for surface waves Boussinesq systems[END_REF], [START_REF] Saut | The Cauchy problem on large time for surface-waves-type Boussinesq systems II[END_REF], [START_REF] Liu | Decay and scattering of small solutions of a generalized Boussinesq equation[END_REF]). Here, the parameters a, b, c, d satisfy the consistency conditions (see [START_REF] Bona | Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media. I : Derivation and Linear Theory[END_REF], [START_REF] Lioyd Bona | Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media : II. The nonlinear theory[END_REF])

a + b + c + d = 1 3 . (4.1.4)
To consider all possible value of the parameters (a, b, c, d) present in the equation is too complicated, here we concentrate our study to the "Hamiltonian generic" case namely the case where b = d > 0, a, c < 0.

In a next paper, we would like to study the Boussinesq equation in the KdV-KdV regime

Statement of the main theorem

(b = d = 0). It is worth saying that to prove a Birkhoff normal form result for this case is much more complicated since in this case the nonlinearity is unbounded and its tame property is failed. However, a recent paper [START_REF] Bernier | Long time dynamics for generalized Korteweg-de Vries and Benjamin-Ono equations[END_REF] gives tools to deal with this problem.

Acknowledgement : I wish to thank Professor Benoît Grébert for motivating me to publish this paper with numerous suggestions and discussions. The author is partially supported by the grant BeKAM ANR-15-CE40-0001-02, and by the Centre Henri Lebesgue, ANR-11-LABX-0020-01.

Statement of the main theorem

We expand the solution in Fourier variables

u(x) = k∈Z * u k e 2iπkx , η(x) = k∈Z * η k e 2iπkx (4.2.1) note that u k = u -k , η k = η -k since u and η are real, then (4.1.1) reads    ∂ t η k = -i2πk 1+4π 2 bk 2 ((1 -4π 2 ak 2 )u k + j+l=k u j η l ) ∂ t u k = -i2πk 1+4π 2 bk 2 ((1 -4π 2 ck 2 )η k + 1 2 j+l=k u j u l ) k ∈ Z * . (4.2.2)
Simplify, we denote

D k = 2πk 1 + 4bπ 2 k 2 , ω k = (1 -4aπ 2 k 2 )(1 -4cπ 2 k 2 ) Ω k = D k ω k .
Before we state our result, we need to put the system (4.2.2) into a more convenient form by using a linear symplectic change of variables

ψ k = 1 √ 2 (α k u k + α -1 k η k ), φ k = 1 √ 2 (α k u k -α -1 k η k )
where

α k = 1-4aπ 2 k 2 1-4cπ 2 k 2 1 4 = α -k .
Then the system can be written as

     ∂ t ψ k = -iD k ∇ ψ -k H ∂ t φ k = iD k ∇ φ -k H k ∈ Z * , ( 4
H 0 = k∈N * ω k (ψ k ψ -k + φ k φ -k ) P = 1 4 √ 2 j+l+h=0 α h α -1 j α -1 l (ψ j + φ j )(ψ l + φ l )(φ h -ψ h )
Through all the paper, we identify the couple function (ψ, φ) with its series of Fourier coefficients (ψ, φ) = (ψ k , φ k ) k∈Z * . We also define the Sobolev space (s ≥ 0)

H s := {z = (ψ, φ) = (ψ k , φ k ) k∈Z * ∈ 2 (T) × 2 (T) | z 2 s = k∈Z * |k| 2s (|φ k | 2 + |ψ k | 2 ) < ∞}. (4.2.4) Denote by N k = |(ψ k , φ k )| 2 = |φ k | 2 + |ψ k | 2
the actions of the flow (ψ, φ)(t). We also denote

I k = |φ k | 2 , J k = |ψ k | 2 then N k = I k + J k = α 2 |u k | 2 + α -2 |η k | 2 . Denote by B s (R)
the open ball centered at the origin of radius R in P s .

Poisson bracket

In the phase space H s , we define the Poisson bracket

{P, Q} = k∈Z * D k ∂ ψ k P ∂ ψ -k Q - k∈Z * D k ∂ φ k P ∂ φ -k Q = k∈N * D k (∂ ψ k P ∂ ψ -k Q -∂ ψ -k P ∂ ψ k Q) - k∈N * D k (∂ φ k P ∂ φ -k Q -∂ φ -k P ∂ φ k Q).
so that the equation (4.2.3) reads In the generic case, since a

     i∂ t ψ k = {ψ k , H} i∂ t φ k = {φ k , H} k ∈ Z * . ( 4
+ c + 2b = 1 3 , we have b = 1 6 -(a + c)/2 > 1 6 and 0 < p = 16π 4 ac ≤ 16π 4 (b - 1 6 ) 2 e = -4π 2 (a + c) = 4π 2 (2b - 1 3 ) ≥ 0.
We denote I b := (0, p 0 = 16π 4 (b -1 6 ) 2 ). In the Kdv-Kdv case, in order for frequencies to be real, we assume a, c > 1 4π 2 . Since a + c = 1 3 , we have 4 3 π 2 -1 < p ≤ 4 9 π 4 .

Scheme of the proof

In order to prove the Theorem 4.2.1, we search iterative changes of variables T n , 1 ≤ n ≤ r, such that

H • T n = H 0 + Z 4 (I, J) + Z 6 (I, J, Y ) + • • • + Z n (I, J, Y ) + R n+1 . (4.2.11)
Where R n+1 is a perturbation term which is of the size µ n+ 3 2 and

I = (I k ) k∈Z * , J = (J k ) k∈Z * , Y = ( k j =0 Y k j ), Y k = ψ k φ k .
Here we call Y k the pseudo actions, which satisfy

{Y k , H 0 } = 0, k ∈ Z * . Moreover {I k + J k , k j =0 Y k j } = 0 ∀k, k j ∈ Z * . (4.2.12)
The condition k j = 0 appear naturally since H comes from an integration of a real function on the circle. As a consequence of the appearance of Y, the exchange of energy mainly occurs between I k and J k for a long time |t| ≤ µ -r+ 3 2 , ∀k ∈ Z * .

Normal form and tame modulus

In this section, we recall some notations and lemmas used in [START_REF] Bambusi | Birkhoff normal form for partial differential equations with tame modulus[END_REF].

Nonresonant condition

Z 6 = k,l∈Z * a k,l Y k Y l Y -k-l + k,l,h∈Z * b k,l,h I k I l I h + c k,l,h I k I l J h + d k,l,h I k J l J h + e k,l,h J k J l J h
where a k,l , b k,l,h , c k,l,h , d k,l,h , e k,l,h are real constants.

Tame modulus

Let us consider a homogeneous polynomial of degree r + 1 f : P s → R, which can be written as f (z) := |j|=r+1 f j z j , j = (. . . , j -l1 , j -l2 , . . . , j -11 , j -12 , j 11 , j 12 , . . . , j l1 , j l2 , . . . )

z j := (ψ, φ) j = . . . ψ j -l1 -l φ j -l2 -l . . . ψ j l1 l φ j l2 l . . . , |j| := l (|j l1 | + |j l2 |),
We can associate f to a symmetric r + 1-linear form f as

f (z) = f (z (1) , . . . , z (r+1) ) = |j|=r+1 f j 1 ,...,j r+1 z (1) j 1 . . . z (r+1) j r+1 , j l = (j l1 , j l2 ).
We say that f is bounded if there exists a constant C such that

|f (z)| ≤ C z r+1 s , ∀z ∈ P s , or equivalently | f (z (1) , . . . , z (r+1) )| ≤ C z (1) s . . . z (r+1) s . Now consider X f := ( 2πk 1+4π 2 bk 2 ∂ ψ k f, -2πk 1+4π 2 bk 2 ∂ φ k f
) the vector field of f. We write it as

X f (z) = l X l (z)e l
where e l is l-th standard basis of P s and X l (z) is a real valued homogeneous polynomial of degree r. We also consider the r-linear form Xf so that X f (z) = Xf (z, z, . . . , z).

Definition 4.3.5. Let f is a homogeneous polynomial of degree r, we define its modulus Lemma 4.3.9. Let f, g ∈ T s,R , then for any d < R we have {f ; g} ∈ T s,R-d and

|{f ; g}| s,R-d ≤ 2 d |f | s,R |g| s,R .
Proof. Write f = j f j and g = k g k with f j homogeneous of degree j and similarly for g. One has

{f, g} = j,k {f j , g k }.
We have estimate for each term of the series

|{f j ; g k }| s,R-d = |{f j , g k }| s (R -d) j+k-3 ≤ 2|f j | s |g k | s (j + k -2)(R -d) j+k-3 ≤ 2|f j | s |g k | s 1 d R j+k-2 = 2 d |f j | s,R |g k | s,R
here we use the estimate

k(R -d) k-1 < R k d , ∀0 < d < R.
Then as in lemma 4.14 in [BG06], we deduce Lemma 4.3.10. Let g, χ ∈ T s,R be two analytic functions. Denote g l = {g l-1 , χ} for all l > 0 with g 0 = g. Then for any 0 < d < R, one has g n ∈ T s,R-d and

|g n | s,R-d ≤ |g| s,R ( 2e d |χ| s,R ) n . (4.3.6) Proof. Denote δ = d n .
Apply iteratively Lemma 4.3.9, one has

|g l | s,R-δl ≤ 2 δ |g l-1 | s,R-δl-1 |χ| s,R ≤ • • • ≤ 2 n n! ( n d |χ| s,R ) n |g| s,R . ( 4
(R). Fix 0 < d < R assume X χ s,R < d 3 , then for |t| ≤ 1, one has sup z s≤R-d ϕ t χ (z) -z s ≤ X χ s,R (4.3.8) X g•ϕ t χ s,R-d ≤ (1 + 3 d X χ s,R ) X g s,R . (4.3.9)
For the proof see [START_REF] Bambusi | Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations[END_REF] proof of lemma 8.2

Remark 4.3.12. For any positive integer number N, we write z = z + z with z = (φ j , ψ j ) |j|≤N and z = (φ j , ψ j ) |j|≥N . Then if homogeneous polynomial f ∈ T s M of order r + 1 has a zero of order three in z, one has

X f s,R ≤ |f | s ( 1 r r l=1 z (1) 1 . . . z (l-1) 1 z (l) s z (l+1) 1 . . . z (r) 1 ) ≤ |f | s z 1 z r-3 1 z s ≤ |f | s R r N s-1 = |f | s,R N s-1 . (4.3.10)
Here we use estimates

z 1 ≤ z s N s-1 z s ≤ z s , z s ≤ z s z 1 ≤ z s .
For a non homogeneous polynomial f ∈ T s,R of order less or equal r + 2

f = j≤r+2 f j then X f s,R ≤ j≤r+2 X f j s,R ≤ j≤r+2 |f j | s R j-1 N s-1 = r |f | s,R N s-1 . (4.3.11)

Homological equation

In each step of Birkhoff normal form procedure, we need to solve a homological equation

-i{H 0 , χ} + Z = f (4.3.12)
with Z is in (κ, α, N )-normal form. Then after each step, we have the new nonlinearity term of size {Z 4 , χ} + {f, χ}. Then we need to estimate the nonlinearity χ and the normal form Z.

with Z is in (κ, α, N )-normal form with respect to Ω. Then by [START_REF] Bambusi | Birkhoff normal form for partial differential equations with tame modulus[END_REF] if f 0 = P has the tame modulus property and frequencies Ω k satisfy a nonresonant condition, there exists χ solving (4.3.16) and

|X χ | s,R ≤ C N α κ |X f | s,R , |X Z | s,R ≤ C |X f | s,R .
(4.3.17)

In our case, we formally expand f in Taylor series

f (φ, ψ) = j,l f j,l Π k,h ψ j k k φ l h h
and similarly for χ, Z. The equation (4.3.16) becomes

(Ω(j -l))χ j,l + Z j,l = k∈N * (Ω k (j k -j -k -l k + l -k ))χ j,l + Z j,l = f j,l . (4.3.18)
Here we use

{H 0 , ψ j k k φ l h h } = (Ω k j k -Ω h l h )ψ j k k φ l h h , Ω k = -Ω -k .
Then when Ω satisfies the nonresonant condition, we define the solution for the homological equation as following

χ jl := f jl (Ω(j -l))
with |Ω(j -l)| ≥ κ N α Z jl := f jl with |Ω(j -l)| < κ N α .

Boussinesq equation's properties

Nonlinearity

Now consider

P = k 1 +k 2 +k 3 =0 α -1 k 1 α -1 k 2 α k 3 (ψ k 1 + φ k 1 )(ψ k 2 + φ k 2 )(ψ k 3 -φ k 3 ).
We prove that P has tame modulus.

Lemma 4.4.1. The nonlinearity P has tame modulus.

Proof. One has that its modulus is

P = k 1 +k 2 +k 3 =0 |α -1 k 1 α -1 k 2 α k 3 |(ψ k 1 + φ k 1 )(ψ k 2 + φ k 2 )(ψ k 3 + φ k 3 ) and XP = (|D k |∂ φ -k P , |D k |∂ φ -k P ) k∈Z * with |D k |∂ φ k P = |D k |∂ ψ k P = 1 2 √ 2 |D k |   j+l+k=0 1 2 |α k α -1 j α -1 l |(ψ j + φ j )(ψ l + φ l )   + 1 2 √ 2 |D k |   j+l+k=0 |α -1 k α -1 j α l |(ψ j + φ j )(ψ l + φ l )   = 1 2 √ 2 |D k | j+l+k=0 ( 1 2 |α k α -1 j α -1 l | + |α -1 k α -1 j α l |)(ψ j + φ j )(ψ l + φ l ).
In the case b = 0, we have that |D k | = |2πk| 1+4π 2 bk 2 ≤ 1 2π|b| . Since |α k | ≤ 1, we verify s-tame map property of X P as

X P 2 s = 1 4 k∈Z * k 2s |D k | 2 ( j+l+k=0 ( 1 2 |α k α -1 j α -1 l | + |α -1 k α -1 j α l |)(ψ j + φ j )(ψ l + φ l )) 2 ≤ 9 64π 2 b 2 k∈Z * k 2s ( j+l+k=0 (ψ j + φ j )(ψ l + φ l )) 2 ≤ 9 16π 2 b 2 z (1) 2
s z (2) 2 1 + z (1) 2 1 z (2) 2 s i.e.

X P s ≤ 3 4π|b| z (1) s z (2) 1 + z (1) 1 z (2) s (4.4.1)

So that P has tame modulus.

Frequencies

We now consider the frequencies The parameter s and p are well-defined, positive and independent. In this subsection, we study frequencies Ω k as functions of p. More precisely, the goal of this part is to prove the following theorem, which has been demonstrated in slightly different contexts in [START_REF] Bambusi | Birkhoff normal form for some nonlinear PDEs[END_REF], [START_REF] Bambusi | Birkhoff normal form for partial differential equations with tame modulus[END_REF] and [START_REF] L Hakan Eliasson | KAM for the nonlinear beam equation[END_REF].

Ω k = 2π k 2 + 4π 2 bk 2 (1 -4aπ 2 k 2 )(1 -4cπ 2 k 2 ) = 2π
Theorem 4.4.2. There exists a set J ∈ I of full measure such that for any p ∈ J , fixed r ≥ 1, there exist κ = κ(r, p) > 0, α = α(r, p) > 0 such that for any N large enough then In order to prove Theorem 4.4.2, we need to construct some lemmas. Here x j = j 4 pj 4 +ej 2 +1 .

| j≥1 Ω j k j | ≥ κ N α , ( 4
Proof. We compute the iterative derivatives of Ω k with respect to p in the spirit of section 3 in [START_REF] L Hakan Eliasson | KAM for the nonlinear beam equation[END_REF] d j Ω k dp j = 2π k 1 + 4π 2 bk 2 (2j -1)! 2 j-1 (j -1)!2 j (-1) j k 4j (pk 4 + ek 2 + 1) j-1 2 = (-1) j (2j -1)! 2 j-1 (j -1)!2 j Ω k x j k .

Boussinesq equation's properties

Substituting this into the determinant then

D = ±   r j=1 (2j -3)! 2 j-2 (j -2)!2 j   r =1 Ω j × 1 1 1 . . . 1
x j 1 x j 2 x j 3 . . . x jr x 2 Proof for Theorem 4.4.2. Define J := κ>0 J κ then it satisfies the Theorem.

Appendix A

transformation T and its inverse fulfill the estimate (4.2.10).

Appendix A

Proof of lemma 4.3.4. Denote X := X f , Y := X g . One has

X = k,l 1 ,••• ,ln D k X l 1 ,••• ,ln k z l 1 • • • z ln e k (4.6.1) Y = k,j 1 ,••• ,jm D k Y j 1 ,•
|nD k D ln X l 1 ,••• ,ln k Y j 1 ,••• ,jm ln ± mD k D ln X l 1 ,••• ,l n-1 ,jm ln Y j 1 ,••• ,j m-1 ,ln k | × e k z l 1 • • • z l n-1 z j 1 • • • z jn .
One has

X {f,g} (z) 2 s ≤ 2 k,l 1 ,••• ,ln j 1 ,••• ,jm (|nD k D ln X l 1 ,••• ,ln k Y j 1 ,••• ,jm ln | 2 + |mD k D ln X l 1 ,••• ,l n-1 ,jm ln Y j 1 ,••• ,j m-1 ,ln k | 2 ) k 2s × (z l 1 • • • z l n-1 z j 1 • • • z jn ) 2 ≤ 2n 2 k,l 1 ,••• ,ln (|D k X l 1 ,••• ,ln k | 2 k 2s (z l 1 • • • z l n-1 ) 2 ln,j 1 ,••• ,jm |D ln Y j 1 ,••• ,jm ln | 2 (z j 1 • • • z jn ) 2 + 2m 2 jm,l 1 ,••• ,ln (|D ln X l 1 ,••• ,l n-1 ,jm ln (z l 1 • • • z l n-1 ) 2 × k,ln,j 1 ,••• ,j m-1 |D k Y j 1 ,••• ,j m-1 ,ln k | 2 (z j 1 • • • z jn ) 2 | 2 k 2s ≤ 2n 2 |f | 2 s |g| 2 0 ( 1 n z l 1 2 1 • • • z l i 2 1 z l i 2 s z l i+1 2 1 • • • z ln 2 1 ) z j 1 2 1 • • • z jm 2 1 + 2m 2 |f | 2 0 |g| 2 s ( 1 m z j 1 2 1 • • • z j i 2 1 z j i 2 s z j i+1 2 1 • • • z ln 2 1 ) z j 1 2 1 • • • z jm 2 1 ≤2(n + m) 2 |f | 2 s |g| 2 s ( 1 n + m -1 z l 1 2 1 • • • z l i 2 1 z l i 2 s z l i+1 2 1 • • • z l n+m-1 2 1 ).

Appendix B

In this section, we prove the local well-posedness for the Boussinesq system (see chapter 16 [Tay3]).

The case b = 0

Existence By the change of variables, the well-posedness of (4.2.3) implies the well-posedness of (4.1.1). The idea here is to obtain a solution to (4.1.1) as a limit of solutions (ψ ε , φ ε ) to Where ϕ ∈ C ∞ 0 (R) is an even, real valued satisfying ϕ(x) = 1 for any x ∈ [-1, 1]. Since ϕ has support bounded in R, the sum (4.7.2) and (4.7.3) are finite. As a consequence, the system (4.7.7) is of finite ODEs. By Cauchy-Lipschitz theorem, we know that there is a unique solution to this system, for t close to 0. First, we will prove that the solution (ψ ε , φ ε )(t) exists in an interval [-T, T ] independent of ε ∈ (0, 1]. Lemma 4.7.1. Let s ≥ 1, 0 < (ψ ε , φ ε )(0) s = µ, then (ψ ε , φ ε )(t) s ≤ 2µ ∀t ∈ [-bπ 3µ , bπ 3µ ] independent of ε ∈ (0, 1].

     ∂ t ψ ε,k = -iD k ∇ ψ ε,-k H(ψ ε , φ ε ) ∂ t φ ε,k = iD k ∇ φ ε,-k H(ψ ε , φ ε ).
Proof. Since P has tame modulus one has an estimate

∂ t (ψ ε , φ ε )(t) 2 s = |{ (ψ ε , φ ε )(t) 2 s , P }| ≤ (ψ ε , φ ε )(t) s X P (ψ ε , φ ε ) s ≤ 3 2πb (ψ ε , φ ε )(t) 2 s (ψ ε , φ ε )(t) 1 .
By Gronwall's inequality, one has that (ψ ε , φ ε )(t) s ≤ 2µ for all t ∈ [0, bπ 3µ ], where we assume (ψ(x, 0), φ(x, 0)) s = µ 1. In the same way, we also have (ψ ε , φ ε )(-t) s ≤

Appendix B

2µ for all t ∈ [0, bπ 3µ ]. Denote I = [-bπ 3µ , bπ 3µ ], then (ψ ε , φ ε )(t) is uniformly bounded in C(I, H s ) ∩ C 1 (I, H s-1 ). Lemma 4.7.2. Let s ≥ 1, the sequence {(ψ ε , φ ε )(t)} ε∈(0,1] is Cauchy in H s for all t ∈ I.

Proof. We consider (v, w)(t) = (ψ ε 1 , φ ε 1 )(t) -(ψ ε 2 , φ ε 2 )(t). Then for any N > 0, there exists ε N ∈ I such that for any ε 1 , ε 2 ≤ ε N , one has

(ψ ε 1 , φ ε 1 )(0) -(ψ ε 2 , φ ε 2 )(0) s ≤ 1 N s µ.
The key point here is (ψ ε 1 , φ ε 1 ) k (0) = (ψ ε 2 , φ ε 2 ) k (0) for all |k| ≤ N. Moreover (v, w)(t) satisfies equation

     ∂ t v k = -iD k (∇ ψ ε 1 ,-k H(ψ ε 1 , φ ε 1 ) -∇ ψ ε 2 ,-k H(ψ ε 2 , φ ε 2 )) ∂ t w k = iD k (∇ φ ε 1 ,-k H(ψ ε 1 , φ ε 1 ) -∇ φ ε 2 ,-k H(ψ ε 2 , φ ε 2 )). ( 4

Uniqueness

Assume that (ψ (1) , φ (1) ), (ψ (2) , φ (2) ) are two solutions of (4.2.3) satisfying (ψ (1) , φ (1) )(x, 0) = (ψ (2) , φ (2) )(x, 0). Denote (ν, υ) = (ψ (1) , φ (1) ) -(ψ (2) , φ (2) ), then by computation one has

The case b = 0

In this case, since D k = 2πk is not bounded, we do not have tame property, so that the proof for existence and uniqueness of (ψ, φ) solution has to be modified. As in the case b = 0, we define (4.7.7)

u ε (t) =

Appendix B

Then one has

∂ t k∈N * |k| 2s (α 2 k |η ε,k | 2 ) + α -2 k |u ε,k | 2 ) = 2π 1 2 j+l+k=0 (α 2 l l 2s+1 + α 2 k k 2s+1 )u ε,j η ε,l η ε,k
+ 2π 1 6 j+l+k=0 (α -2 l l 2s+1 + α -2 k k 2s+1 + α -2 j j 2s+1 )u ε,j u ε,l u ε,k .

Here α 2 k = ( 1-4π 2 ak 2 1-4π 2 ck 2 ) 1/2 . One has

α 2 k = a c + O( 1 k ); α -2 k = c a + O( 1 k ).
So that Lemma 4.7.6. Let s ≥ 1, the sequence {(ψ ε , φ ε )(t)} ε∈(0,1] is Cauchy in H s for all t ∈ I.

|α 2 l l 2s+1 + α 2 k k 2s+1 |
Proof. We consider (v, w)(t) = (ψ ε 1 , φ ε 1 )(t) -(ψ ε 2 , φ ε 2 )(t). Then for any N > 0, there exists ε N ∈ I such that for any ε 1 , ε 2 ≤ ε N , one has Here u ε,l = u ε 1 ,l + u ε 2 ,l and η ε,l = η ε 1 ,l + η ε 2 ,l , ∀l ∈ Z * . By Gronwall's inequality, one has that (v, w)(t) s ≤ 4 µ N s for all t ∈ I.

(ψ ε 1 , φ ε 1 )(0) -(ψ ε 2 , φ ε 2 )(0) s ≤ 1 N s µ.
As in the case b = 0, one has lim ε→0 (u ε , η ε ) → (u, η) ∈ C(I, H s ) solution of Boussinesq systems. Similarly, we have the uniqueness for (u, η).

  f, g, h in the phase space and time, one has properties of the Poisson bracket -Anticommutativity {f, g} = -{g, f }, -Bilinearity {af + bg, h} = a{f, h} + b{g, h} a, b ∈ R, -Leibniz's rule {f g, h} = {f, h}g + f {g, h}, -Jacobi identity {f, {g, h}} + {g, {h, f }} + {h, {f, g}} = 0. Moreover, we have an equivalent expression of the Poisson bracket of functions to the Lie bracket of the associated Hamiltonian vector fields X {f,g} = -[X f , X g ]. A Hamiltonian equation can be written in term of the Poisson bracket u = {u, H} (1.1.1)

Introduction

  and a symplectic transformation ΦS = Φ t S | t=1 , one has {f, g} • Φ S = {f • Φ S , g • Φ S }.

  2π periodic potential, having average m, and g ∈ C ∞ (T × U), U being a neighbourhood of the origin in R. It is proved in [Bam03 ; BG06] that for a large set of m, the frequencies ω j = |j 2 | + m of the linear wave operator satisfy a nonresonant Introduction condition. Precisely, fix r ≥ 3, there exist γ > 0 and α = α(r) such that

  [BCS02 ; BCS04]. In fact, different possible values of a, b, c, d form different Boussinesq systems with different properties, such systems are specialized as subclasses : classical Boussinesq system (a = b

Introduction

  the time in averaged operators.

  action at mode k. One has the following Birkhoff normal form result Theorem 1.2.6. Let r ≥ 1, s ∈ R sufficiently large and 0 < µ 1, then there exists a subset I µ ⊂ I b asymptotically of full measure, and a constant C = C(r, s, b) such that for any p ∈ I µ , for |t| ≤ µ -r+3/2 , there exists a transformation T : B s (µ/3) → B s (µ) satisfying

  are the actions corresponding to the internal modes (r, θ) ∈ (R n × T n , dr ∧ dθ) ; -L and F are respectively infinite and finite sets, Z is the disjoint union L ∪ F; ζ = (ζ a ) a∈Z ∈ C Z are the external modes endowed with the standard complex symplectic structure -idζ ∧ dη. The external modes decomposes in a infinite part ζ L = (ζ a ) a∈L , corresponding to elliptic directions, which means Λ a ∈ R for a ∈ L, and a finite part ζ F = (ζ a ) a∈F , corresponding to hyperbolic directions, which means Im Λ a = 0 for a ∈ F; -L has a clustering structure L = ∪ j∈N L j , where L j are finite sets of cardinality

  transformation, and analytically depending on ρ ∈ D, satisfying -Φ ρ : O s (σ, µ) → T ρ (ν, 2σ, 4µ, s); -Φ ρ puts the Hamiltonian h in normal form in the following sense :

  2.11. It is crucial to obtain (3.4.38) and (3.4.39) which express the control of the new remainder R + after one KAM step in two different norms, a low s-decay norm and a high s + b-decay norm. The parameter N measures the troncature in the Fourier variable associate to the angle ϕ = ωt and in the off-diagonal distance in the matrix (see (3.4.20)). When iterating the procedure, this special form of estimates (3.4.38)-(3.4.39) allows to obtain a convergent scheme for the sequence of remainders R k when choosing conveniently the sequence of troncature parameter N k .

  r with the family of semi-norms N m,r,p (A) 2 := l∈Z d l 2r N 2 m,p (A(l)) , p ≥ 1 . (3.2.31) Consider a Lipschitz family O ω → A(ω) ∈ A m,r where O is a compact subset of R d , d ≥ 1. For γ > 0 we define the Lipschitz semi-norms (recall (3.2.16)) as N γ,O m,r,p (A) := N sup,O m,r,p (A) + γN lip,O m,r,p (A) (3.2.32) We denote by A γ,O m,r the Fréchet space of families of pseudo differential operators A(ω) ∈ A m,r endowed with with the family of semi-norms {N γ,O m,r,p } p≥1 . Similarly we define the corresponding class of ρ-smoothing operators R(ω, ϕ), H r in ϕ and Lipschitz in ω. Definition 3.2.4. Let ρ ∈ R and r > d/2. We denote by R ρ,r the Fréchet space of ρsmoothing H r -mapping T d ϕ → R(ϕ) ∈ L(H s , H s+ρ ) for all s ∈ R endowed with the family of semi-norms |R| 2 ρ,r,s := l∈Z d l 2r R(l) 2 L(H s ,H s+ρ ) , s ∈ R . (3.2.33) Consider a family O ω → R(ω) ∈ R ρ,r where O is a compact subset of R d , d ≥ 1. For γ > 0 we denote by R γ,O ρ,r the Fréchet space of families of pseudo differential operators R(ω) ∈ R ρ,r endowed with with the family of semi-norms {N γ,O ρ,r,p } p∈N defined by (recall (3.2.16)) |R| γ,O ρ,r,p := |R| sup,O ρ,r,p + γ|R| lip,O ρ,r,p . (3.2.34) We notice that by (3.2.21) we have A m,r ⊂ R -m,r . Lemma 3.2.5. Let r > d/2, m, ρ ∈ R and consider R ∈ R γ,O ρ,r and A ∈ A γ,O m,r . Then, for any s ∈ R, there are C = C(s, r) > 0, p(s, m) > 0 such that

  Let s ∈ [s 0 , Sb]. There exist positive constants 0 = 0 (s), C = C(s) such that, if ≤ 0 , (3.4.6) then there is a set O ⊆ O with meas(O \ O ) ≤ Cγ (3.4.7) such that the following holds. For any ω ∈ O there are (i) (Normal form) a matrix Z ∞ = Z 0 + Z∞ with Z∞ ∈ M γ,O ρ,s which is ϕ-independent, Hermitian and block-diagonal ; (ii) (Conjugacy) a bounded and invertible map

  Assume that [[Z 2 ]] γ,O κ,s 0 +b ≤ γ/8 (3.4.14) Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential for some 0 < γ ≤ c 0 5 (see (3.2.6)) then we have meas O \ O + (γ, N ) ≤ CγN -1 (3.4.15)

)

  Lemma 3.4.4. (Homological equation). Let R ∈ M γ,O ρ,s for s ∈ [s 0 , S], ρ in (3.4.2). For any ω ∈ O + ≡ O + (γ, N ) (defined in (3.4.13)) there exist Hermitian matrices S, Q solving equation (3.4.16) and satisfying

  Lemma 3.6.4, and since the regularity in ϕ has been fixed at r = b, one deduces the estimates (3.4.19). Moreover, recalling (3.4.17), we have that equation (3.4.16) is Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential equivalent toG(l, k, k , ω)S [k ] [k] (l) + (Π N R) l ∈ Z d , k, k ∈ N with (l, k, k ) = (0, k, k)where the operator G(l, k, k , ω) is the linear operator acting on complex d k × d k -matrices as

  .4.26) Then, by denoting by • ∞ the sup-norm of a d k × d k -matrix, we deduce

  for N 0 large enough where we used (3.4.43) and b ≥ a+2 . The (3.4.58) is the first estimate in (3.4.49) at step 1, the other is proved similarly. Proof of Theorem 3.4.1. Consider the operator L 0 in (3.4.1). The smallness condition (3.4.6) implies the (3.4.43), hence Proposition 3.4.8 applies. We define the set O ≡ O ∞ := ∩ ν≥0 O ν . (3.4.59) By the measure estimate (3.4.50) we deduce (3.4.7). For any ω ∈ O ∞ , ν ≥ 0, we define (see (3.4.45), (3.4.46)) the map

  .4.66) Now fix s ∈ [s 0 , Sb], since by hypothesis (A2), R 0 ∈ M γ,O ρ,s+b , we deduce from the last estimate that ( Ψ ν ) ν≥0 is a Cauchy sequence in M γ,O∞ s . Hence Φ ν →Φ ∞ ∈ M γ,O∞ s . Furthermore by (3.4.66) one deduces the (3.4.9). The estimate on Φ -1 ∞ -Id follows by using Neumann series and reasoning as in the proof of Lemma 3.6.3. By (3.4.47) we deduce that Z ν,2 is a Cauchy sequence in M γ,O∞ ρ,s . Hence we set

Chapitre 3 -

 3 Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential by (3.4.5) satisfies≤ N δ,r,p (W )ε 1-α (3.5.4)

  .5.5) for any (d + n)/2 < s < Sb, such thatL 0 ≡ F + transforms into L ∞ in (3.4.8). By (3.4.7) we have meas(O 0 \ O ) ≤ Cγ , for some constant C > 0 depending on s. It is also know that (recall (3.3.2)) meas([1/2, 3/2] d \ O 0 ) ≤ Cγ.Therefore, recalling that we set γ = ε α we have that the (3.1.2) holds. Moreo-ver, by Lemma 3.6.2 and (3.5.5) we have supϕ∈T d Φ ±1 ∞ (ϕ) -Id L(H s ;H s ) S N δ,r,p (W )ε 1-α .(3.5.6)For ω ∈ O ε we setΨ(ωt) := Φ ∞ (ωt) • Φ(ωt) .By construction the function v := Ψ(ωt)u satisfies the equation (3.1.5) with εZ Z ∞ in (3.4.8). Moreover, by (3.3.8), (3.3.9), (3.5.6) and (3.5.2), we have sup

  .2.5) Frequencies Let us write Ω k = 2πk 1+4bπ 2 k 2 √ pk 4 + ek 2 + 1 where p = 16π 4 ac, e = -4π 2 (a + c). Here we assume that b = d ≥ 0, and we consider two different cases : the "generic" case b = d > 0, a, c < 0 and the Kdv-Kdv case b = d = 0, a, c > 0.

k 1 +

 1 4π 2 bk 2 pk 4 + ek 2 + 1 where e = -4π 2 (a + c) > 0, p = 16π 4 ac ∈ (0, p 0 = 16π 4 ( 1 6 -b) 2 ) = I b .

  .4.2)for any k ∈ Z ∞ , fulfilling 0 = |k| = j |k j | ≤ r + 2, |j|>N |k j | ≤ 2.

Lemma 4.4. 3 .

 3 For any r ≤ N, consider r indexes j 1 < j 2 < j 3 < ...j r ≤ N ; consider the determinant

k 2 .

 2 2 bj 2 pj 4 + ej 2 + 1 j -x j k )| = 1≤ <k≤r |j 2 + j 2 k + ej 2 j 2 k ||j 2 -j 2 k | (pj 4 + ej 2 + 1)(pj 4 k + ej 2 k + 1) ≥ 1≤ <k≤r 2(|j | + |j k |) (p + e + 1) 2 j 3 j 3 k ≥ 1≤ <k≤r 2(|j | + |j k |) (4π 2 (b -1 6 ) + 1) 4 j 3 j 3 <k≤r 2(|j | + |j k |) (4π 2 (b -1 6 ) + 1) 4 j 3 j 3 r-1)/2 (4π 2 (b -1 6 ) + 1)2r(r-1) . Now we recall a result from [BGG85] appendix B Lemma 4.4.4. Let u (1), • • • , u (r) be r independent vectors in R r of norm at most one, Chapitre 4 -Birkhoff normal form for abcd Boussinesq system on the circle and let w ∈ R r be any non-zero vector. Then there existsi ∈ [1, • • • , r] such that | u (i) , w | ≥ |w||det(u (1) , • • • , u (r) )| r 3By[E02] (see also [EGK16 ; XYQ97]), we have the lemma Lemma 4.4.5. Suppose that g(p) be a C r -smooth function on an intervalJ ⊂ R such that |g (p)| C r-1 ≤ β. Let J h := {p ∈ J : |g(p)| < h}, h > 0. If max 1≤k≤r min p |g (k) (p)| ≥ d then |J h | ≤ C r ( β h + 1)( h d ) 1/r . Now let us consider the function g(p) = |k| -1 ( N |a|=1 k a Ω a + c), with |k| ≤ r < N , c is a fixed constant, then |g (p)| C n ≤ C(n, b). Define R kc (κ, α) := {p ∈ I b | |g(p)| ≤ κ |k|N α } By lemma 4.4.4 one has max 1≤k≤r min p |∂ k g(p)| ≥ C(r, b)N -1 2 (5r 2 -3r)-2 so that |R kc (κ, α)| ≤ 2 C(r, b) κ 1/r N τ with τ = 1 2 (5r -3r) + 4 -α r .Lemma 4.4.6. Fix α > 5 2 r 3 + r 2 + 7r, and let I κ = I \ kc R c,k (κ, α) then for any p ∈ I κ and for any c ∈ βZ, k ∈ Z N with 0 = |k| ≤ r, β > 0, one has| N |a|=1 k a Ω a + c| ≥ κ N α Moreover |I -I κ | ≤ Cκ 1 r where C = 2 √ p+e+1 C(r,b)πb .Proof. Since|Ω k | = | 2πk 1 + 4π 2 bk 2 pk 4 + ek 2 + 1| ≤ √ p + e + 1 2πb |k| We have that | N |a|=1 k a Ω a + c| ≥ 1 for all |c| ≥ √ p+e+1 2πb rN. hence | kc R c,k (κ, α)| ≤ |k|≤r,|c|≤CrN |R c,k (κ, α)| For any 0 < κ 1, there exist α > 0 and a set J κ satisfying |I -J κ | ≤ Cκ α+1 r+2Such that ∀p ∈ J κ one has|Ω N • k + e 1 Ω j + e 2 Ω l | ≥ κ N α (4.4.3) for any k ∈ Z N , |e i | ≤ 1, |j| ≥ |l| > N, and |k| + |e 1 | + |e 2 | = 0, |k| ≤ r + 2.Proof. One has an estimate for Ω j (and Ω l also)Ω j = 2π √ ac b j + a j with |a j | ≤ C j .Hencee 1 Ω j + e 2 Ω l = 2π√ ac b (e 1 j + e 2 l) + e 1 a j + e 2 a l .If j, l > CN α /κ then e 1 a j + e 2 a l is just an irrelevant terms, so that the estimate (4.4.3) follows from Lemma 4.4.6 with β = 2π√ ac b . For the case j, l ≤ CN α /κ, we reapplies the lemma with N, r replaced by N = CN α /κ and r = r + 2. Then the nonresonance condition (4.4.3) holds provided α = α 2 ≈ r 6 and the complement set is measured by a constant times κ α+1 r+2 .

  (4.7.1)Here (ψ ε , φ ε ) is defined by ψ ε (t) = k∈Z * ϕ(εk)ψ k (t)e i2πkx(4.7.2)φ ε (t) = k∈Z * ϕ(εk)φ k (t)e i2πkx .(4.7.3)

  υ)(x, 0) = 0 ∂ t (ν, υ) 2 s ≤ C (ν, υ) s (|(ψ (1) , φ (1) ) 2 s + (ψ (2) , φ (2) ) 2 s ) (4.7.6)for some constant C. By Gronwall's inequality it is proved that (ν, υ) ≡ 0. So that the solution (ψ, φ) is unique.Finally we have proved Proposition 4.7.5. Let s ≥ 1, b > 0 the system (4.1.1) is locally well-posed for sufficiently small initial datum in H s .

∂

  k∈Z ϕ(εk)u ε,k (t)e i2πkx u ε,k (0) = u k (0) η ε (t) = k∈Z ϕ(εk)η ε,k (t)e i2πkx η ε,k (0) = η k (0t η ε,k = -i2πk((1 -4π 2 ak 2 )u ε,k + j+l=k u ε,j η ε,l ) ∂ t u ε,k = -i2πk((1 -4π 2 ck 2 )η ε,k + 1 2 j+l=k u ε,j u ε,l ).

Proposition 4.7. 7 .

 7 Let s ≥ 1, b = 0 the system (4.1.1) is locally well-posed for sufficiently small initial datum in H s . Titre : titre (en français) Trois résultats sous forme normale pour les équations de Schrödinger et le système abcd de Boussinesq Mot clés : Théorie KAM, forme normale de Birkhoff, réductibilité, l'équation de Hamilton ,l'équation de Schrödinger,le système abcd de type Boussinesq Résumé : On montre des résultats de forme normale pour des EDPs Hamiltoniennes : l'équation de Schrödinger non linéaire quintique sur le cercle, l'équation de Schrödinger sur une variété Zoll et le système abcd de type Boussinesq sur le cercle. Ces résultats sont démontrés à l'aide de procédure KAM et de procédure de forme normale de Birkhoff. On déduit des résultats de forme normale le comportement en temps long des solutions au voisinage de zéro. Title: titre (en anglais) Three normal form results for Schrödinger equations and abcd Boussinesq system Keywords: KAM theory, Birkhoff normal form, Reducibility, Hamiltonian equations, Schrödinger equation, abcd Boussinesq system Abstract: We prove normal form results for Hamiltonian PDEs: the quintic nonlinear Schrödinger equation on the circle, the Schrödinger equation on a Zoll manifold and the abcd Boussinesq system on the circle. These results are proved via KAM procedure and Birkhoff normal form procedure. As corollaries of normal form results, one deduces the long time behavior of solutions near to zero.

  denotes the third largest number among the collection (|k i |, | j |) i,j .

	The condition (*) appears not only in the case of one dimensional wave equations but also
	in many different contexts, such as the one dimensional nonlinear Schrödinger equation
	with external potential [Bam03 ; Bam08 ; BG06] or even in multidimensional space for
	nonlinear Schrödinger equations [BG06 ; FGL13], and wave equations on Zoll manifolds
	[BDGS07].

  There are many works concerning the theory of existence, uniqueness and long-time dynamical behaviors as well as the regularity problems of the NLS equations. In R d , we quote

	Introduction
	[BGT80 ; GV92 ; G00 ; K87 ; Y87 ; KT98] for the energy subcritical problems, and [Bou99 ;
	CKSTT08 ; KM06 ; KM10 ; TVZ07 ; TVZ08] for the energy critical cases. For the NLS
	equations in R d , the Morawetz's inequalities and Strichartz estimates play a crucial role.
	The Cauchy problem for the nonlinear Schrödinger equation on torus T d was studied by
	Bourgain [Bou93bb], where he extended the classical Strichartz's inequalities to T d in all
	dimensions. These inequalities are called moment estimate for trigonometric polynomials.
	In this paper, Bourgain proved that
	-(d = 1). The NLS equation is locally well-posed for ψ ∈ H s (T), provided p <
	2 + 2 1-2s ;
	-(d = 2). The NLS equation is globally well-posed for p = 4 with initial data in
	H 1 (T 2 ) and sufficiently small L 2 -norm. The same result holds for all α ≥ 2 for
	sufficiently small H 1 -data ;
	-(d = 3). The NLS equation is globally well-posed for 4 ≤ p < 6 with sufficiently
	small initial data in H 1 (T 3 );
	-(d ≥ 4). The NLS equation is locally well-posed for 4 ≤ p < 2 + 4 d-2s and s > 3d d+4 .
	See also [Bou93aa ; Bou93bb ; Bou13]. In general compact manifolds, the approach to
	the Strichartz estimates is much different from Bourgain. The Cauchy problem of NLS
	on general compact manifolds was initiated by Burq et al [BGT02 ; BGT04 ; BGT05 ;
	BGT09].
	in H 1 , one has the following cases
	-Critical (p = 2 + 4 d-2 ) : blowup can occur or global solution can exist.
	-Subcritical (p < 2 + 4 d-2 ) : global solutions exist.

  ≤ s and any ω ∈ O ε the map t → u(t, •) ∈ H s (M n ) solves (3.1.1) if and only if the map t → v(t, •) := Ψ(ωt)u(t, •) solves the autonomous equation

	.1.4)
	• for any n 2 < s

  Let r ≥ 5d/2 + 9/2 and ω ∈ O 0 (see (3.3.2)). Then the operator T in (3.3.27) belongs to A γ,O 0 δ ,r-(2τ +1) is Hermitian, commutes with the operator K 0 . Moreover it solves the equation

	3.3. Regularization procedure
	Lemma 3.3.3.

  By using the expansions (3.2.47)-(3.2.48) and since T solves (3.3.28) we have that the conjugate L 1 has the form (3.3.32)-(3.3.33) with W 1 as in (3.3.34) and iA 1

  Birkhoff normal form for abcd Boussinesq system on the circle with

	Chapitre 4 -
	.2.3)
	where H is the Hamiltonian
	H = H 0 + P
	123

  If the frequency Ω is nonresonant then any polynomial which is in (κ, α, N )-normal form depends only on I, J and Y.

	Remark 4.3.3. Example 4.3.4. If the frequency Ω is nonresonant then a polynomial of order 6, which
	is in (κ, α, N )-normal form with respect to Ω would be
	|Ω(k -l)| <	κ N α and	|j|≥N +1	|k j | + |l j | ≤ 2.
	Definition 4.3.2 (Nonresonant condition). Let r be a positive integer, we say that the
	frequency Ω is nonresonant if there exist κ > 0, and α ∈ R such that for any N large
	enough one has
		|	j∈N *	Ω j k j | ≥	κ N α	(4.3.1)

Definition 4.3.1. Fix two positive parameters κ and α, and a positive integer N. A function Z is said to be in (κ, α, N )-normal form with respect to Ω if Z kl = 0 implies for any k ∈ Z ∞ , fulfilling 0 = |k| := j |k j | ≤ r + 2, |j|>N |k j | ≤ 2.

  .3.7)Using the inequality n n < N !e n , one has the thesis.

Lemma 4.3.11. Let χ, g be two analytic functions with Hamiltonian vector fields analytic Chapitre 4 -Birkhoff normal form for abcd Boussinesq system on the circle in B s

  •• ,jm k z j 1 • • • z jn e k (4.6.2) here we write z l to represent φ l or ψ l . Remind that D k = 2πk 1+4π 2 bk 2 . Then X {f,g} (z) =

k,l 1 ,••• ,ln j 1 ,••• ,jm

  ( (ψ ε 1 , φ ε 1 )(t) 1 + (ψ ε 2 , φ ε 2 )(t) 1 ) Proof. By lemma 4.7.2, the bounded function (ψ ε , φ ε ) converges to (ψ, φ) in L ∞ (I, H s ) when ε → 0. Similarly lim ε→0 X H(ψε,φε) → X H(ψ,φ) in L ∞ (I, H s-1 ), while as a result ∂ t (ψ ε , φ ε ) → ∂ t (ψ, φ) in L ∞ (I, H s-1 ). Thus (ψ, φ) ∈ L ∞ (I, H s ) ∩ Birkhoff normalform for abcd Boussinesq system on the circleProof. It is enough to prove (ψ, φ)(t) s is continuous of t. Similar as lemma 4.7.1, one has so that (ψ, φ) ∈ C(I, H s ) and (ψ, φ)(t) s ≤ 2µ for all t ∈ I.

	∂ t (ψ, φ)(t) 2 s ≤	3 2πb	(ψ, φ)(t) 2 s (ψ, φ)(t) 1
	≤ s + 3 2πb (v, w)(t) 2 3 2πb (v, w)(t) ∀s ≥ 1.	(4.7.5)
	Lip(I, H s-1 ) is a solution of (4.2.3).

.7.4) Then ∂ t (v, w)(t) 2 s = k∈N * k 2s (|∂ t v k v k | + |∂ t w k w k |) ≤ (v, w)(t) s ( X P ((v, w), (ψ ε 1 , φ ε 1 )) s + X P ((v, w), (ψ ε 2 , φ ε 2 )) s ) s (v, w)(t) 1 ( (ψ ε 1 , φ ε 1 )(t) s + (ψ ε 2 , φ ε 2 )(t) s )

By Gronwall's inequality, one has that (ψ ε , φ ε )(t) s ≤ 4 µ N s for all t ∈ I.

Lemma 4.7.3. Let s ≥ 1 and b = 0, then provided the initial datum (ψ ε (0), φ ε (0)) ∈ H s , ε ∈ (0, 1], with (ψ ε (0), φ ε (0)) s = µ small enough, the system (4.2.3) admits a solution (ψ, φ)(t) for all t ∈ I = [ -bπ 3µ , bπ 3µ ] with

(ψ, φ) in L ∞ (I, H s ) ∩ Lip(I, H

s-1 ) Lemma 4.7.4. The solution (ψ, φ)(t) given in Lemma 4.7.3 is in C(I, H s ), ∀s ≥ 1. Chapitre 4 -

  C|j|(|l| 2s + |k| 2s ) Since j + l + k = 0, we could estimate the term |j|(|l| 2s + |k| 2s ) by µ s 1 µ s 2 µ 3 , where µ 1 , µ 2 , µ 3 are the first, the second and the third large term among |j|, |l|, |k|. Similarly, one has|α -2 l l 2s+1 + α -2 k k 2s+1 + α -2 j j 2s+1 | ≤ Cµ s 1 µ s 2 µ 3 .

	≤ ||k 2s+1 | ≤ Then applying the Young inequality, we have a c |l 2s+1 + k 2s+1 | + |α 2 l -a c ||l 2s+1 | + |α 2 k -a c
	|∂ t
	k∈N

* |k| 2s (α 2 k |η ε,k | 2 ) + α -2 k |u ε,k | 2 )| ≤ C (u ε , η ε ) 3 s for some constant C. By Gronwall's inequality, one has that (ψ ε , φ ε )(t) s ≤ 2µ for all t ∈ [0, 2 Cµ ]. Denote I = [-2 Cµ , 2

Cµ ], then (ψ ε , φ ε )(t) is uniformly bounded in C(I, H s ) ∩ C 1 (I, H s-3 ).

  Chapitre 4 -Birkhoff normal form for abcd Boussinesq system on the circlek 2s (|∂ t v k v k | + |∂ t w k w k |) 2s+1 + α 2 k k 2s+1 ||u ε,j |(|η ε,l w k | + |w l ||η ε,k |) (|u ε,j |u ε,l v k | + |u ε,j |u ε,k v l | + |u ε,k |u ε,l v j |) ≤ C (v, w)(t) s ( (ψ ε 1 , φ ε 1 )(t) 2 s + (ψ ε 2 , φ ε 2 )(t) 2 s ).

	Moreover (v, w)(t) satisfies
	∂ t (v, w)(t) 2 s =
	k∈N ≤ 2π 1 2 j+l+k=0 l l + 2π |α 2 1 6 j+l+k=0 |α -2 l l 2s+1 + α -2 k k 2s+1 + α -2 j j 2s+1 |
	×

*

actually [∆ g , Z] = 0 on sphere while on Zoll manifold we have Z and ∆ g can be diagonalized in the same basis of L 2 (M n ).

Here Z * = Z \ {0}

it happens when q -p is odd

in this case, {p, q, s, t} is of the form {p, p + 2n, p + 3n, p -n}

actually [∆ g , Z] = 0 on sphere while on Zoll manifold Z and ∆ g can be diagonalized in the same basis of L 2 (M n ).

In fact the linearization of (3.1.7) gives rise to a system of linear equations for u and u.

Notice that this result holds true for any pseudo-differential perturbation of order δ < 2 depending smoothly on time.

This fact is quite evident in the case of pseudo-differential operators on R n and thus extends to pseudo-differential operators on M n by passing to local charts.

Acknowledgment. During the preparation of this work the three authors benefited from the support of the Centre Henri Lebesgue ANR-11-LABX-0020-01 and of ANR -15-CE40-0001-02 "BEKAM" of the Agence Nationale de la Recherche. R. F. was also supported by ERC starting grant FAFArE of the European Commission and B.G. by ANR-16-CE40-0013 "ISDEEC" of the Agence Nationale de la Recherche.

as in the proof of Lemma 3.3.2 one gets the estimates (3.3.35), (3.3.36). In particular, since δ ≤ 0 we shall use Lemma 3.6.8 in order to estimate the conjugates of smoothing operator under the flow Φ τ T .

In the next Lemma we study the case in which the generator T of Lemma 3.3.3 has order δ > 0.

Lemma 3.3.5. Let 0 < δ ≤ δ. Let us define δ 1 := δ + δ -1 and Φ T := Φ 1 T . Fix moreover r 1 > d/2 and ρ 1 ≥ 0 and assume r > max(r 1 +d/2, 2τ +2+d/2) and ρ ≥ ρ 1 +δ r 1 +1. Then the conjugated operator

T (see (3.3.14)) has the form (3.3.32), (3.3.33), (3.3.34), is independent of

Moreover for any s ∈ R there exist p = p(s, ρ) and if (3.3.13) holds then C = C(s, ρ) such that

Proof. One reasons as in the proof of Lemma 3.3.4. The difference is in estimating the remainder R 1 in (3.3.38). Since the generator T is of order δ > 0 one has to apply Lemma 3.6.7 (instead of Lemma 3.6.8) which provides estimates (3.3.42) instead of the (3.3.36).

Proof of Theorem 3.3.1

In this section we give the proof of Theorem 3.3.1 which is based on an iterative application of Lemmata of the previous section. Recalling (3.3.1) we set

The operator G 0 above has the form (3.3.12) with

(3.3.43)

Since V is C ∞ , r > d/2 can be chosen arbitrary large. We will chose it later in function of the order δ, of the final regularity r 0 and the smoothness ρ 0 prescribed by (3.3.7). Lemma 3.3.2 provides p 1 (S) such that if p ≥ p 1 (S) in (3.3.3) then (3.3.13) holds for any s ∈ [s 0 , S]. By applying Lemma 3.3.2 to G 0 we obtain a symplectic map Φ S 0 such that

KAM reducibility

We perform N = N (ρ 0 , δ) steps of this procedure in order to get δ N = δ -N (1-δ) ≤ -ρ 0 . This require to choose r 1 (and hence r) sufficiently large. More precisely, we want r N ≥ r 0 , the prescribed regularity, and thus in view of (3.3.45) r 1 ≥ N (2τ + 2) + r 0 . Then recalling that we need r > max(r 1 + d/2, 2τ + 2 + d/2) we have to chose

Moreover the constant ρ appearing in (3.3.44) should be chosen in such a way

Therefore the operator G N , defined as in (3.3.46), has the form (3.3.4) with Z := W N . We notice that The case 1/2 ≤ δ < 1 requires to apply Lemmata 3.3.2 and 3.3.5 iteratively to construct Ãn ∈ A δn,rn with δ n = 2δ n-1 -1 and δ 0 = δ, until δn became negative. Then we can apply the second procedure using Lemmata 3.3.2 and 3.3.4 as in the previous case.

KAM reducibility

In this section we will prove an abstract KAM Theorem for a matrix operator of the form

To precise our hypothesis on L 0 we define the following constants

In this section we assume :

(A1) the matrix Z 0 is Hermitian, block diagonal, independent of ϕ and Lipschitz in

[k] , we assume that there exists κ ≥ 0 such that (recall Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential provided that ρ ≥ 5n + 3, which is true by (3.4.2). Combining (3.4.28) and (3.4.31) (recall (3.2.56)) we get the first bound in (3.4.18). The second one follows by (3.6.5) in Lemma 3.6.1. Lemma 3.4.5. There is C(s) > 0 (depending only on s ≥ s 0 ) such that, if

then the map Φ = e iS = Id + Ψ, with S given by Lemma 3.4.4, satisfies

The new remainder

In this subsection we study the conjugate of the operator L under the map Φ given by Lemma 3.4.5. We first define the new normal form Z + as

(3.4.35)

We have the following.

Lemma 3.4.6. (New normal form). We have that Z + in (3.4.35) is ϕ-independent, Hermitian and block-diagonal, and satisfies

where Z + is the normal form given by (3.4.35) and the new remainder R + is Hermitian and satisfies for all s ∈ [s 0 , Sb]

so that the following properties are satisfied for all ν ∈ N :

where Z ν,2 is ϕ-independent, block-diagonal and Hermitian. Moreover it satisfies

Moreover there is a sequence of Lipschitz function

k,j , for j = 1, . . . , d k , are the eigenvalues of the block

where is defined in (3.4.5).

(S3) ν The remainder R ν is Hermitian and satisfies, for any s ∈ [s 0 , Sb],

Proof. We proceed by induction. We first verify the inductive step. So we assume that conditions (Si) j , i = 1, 2, 3, 4, hold for 1 ≤ j ≤ ν. We shall prove that they holds for

k,j . Using the (3.4.47) for s s 0 , we have that

for N 0 ≥ 1 large enough. Hence condition (3.4.14) is satisfied for small enough, i.e.again N 0 large enough (recall (3.4.43)). Therefore Lemma 3.4.3 implies that (3.4.50) holds for the set O ν+1 which is the (S4) ν+1 . We define the new normal form Z ν+1 as (recall (3.4.35))

Lemma 3.4.6, applied with R R ν , together with the estimates (3.4.49), implies the estimate (3.4.47). Let µ

be the eigenvalues of the block (∆+Z ν+1 )

[k]

[k] which are defined on the set O ν+1 . The bound (3.4.48) follows by Lemma 3.4.2 and (3.4.51). Moreover, by Kirtzbraun Theorem, there is an extension µ

to the whole set O 0 with the same Lipschitz norm. This prove the (S2) ν+1 .

Then we want to construct a map Φ ν+1 = Id + Ψ ν+1 . First by the inductive hypothesis (3.4.49) we deduce that (with C(s) given in Lemma 3.4.5)

for small enough and since 2τ + 1 -2 3 a ≤ 0. Hence the smallness condition (3.4.32) holds true. We then apply Lemmata 3.4.4 and 3.4.5 with R R ν and O + O ν+1 and construct a map Φ ν+1 = Id + Ψ ν+1 .

Proof of Theorem 3.1.1

We recall that equation (3.1.1) has the form

and thus W ∈ A δ,r for any r > d/2. Its reducibility rely on the reducibility of the operator F in (3.3.1) with V (ϕ) = εW (ϕ). Rouglhy speaking we want to apply Theorem 3.3.1 to regularize F in such a way operator F is transformed into the operator F + in (3.3.4). Then we apply Lemma 3.2.14 to control the remainder R in (3.3.4) in s-decay norm. This allows, for ε small enough, to apply the reducibility Theorem 3.4.1 and to conclude. To justify all these steps we have to carefully follow the parameters and the smallness conditions. First we fix α ∈ (0, 1) and γ = ε α , δ ≤ 1 2 , s > n/2 and W belonging to all the A δ,r with r > d/2 . Then we fix ρ, b, τ as in (3.4.2), we set κ = 2δ -1 and we fix s 0 > n/2 and S such that s and s + b belong to [s 0 , S] and S ≥ p(δ, 0) (see (3.2.21)). Finally we set

With these values of ρ 0 , r 0 , Theorem 3.

Further we knows that R ∈ R γ,O 0 ρ 0 ,r 0 and

Now we apply Lemma 3.2.14 to conclude that R ∈ M γ,O 0 ρ,S and

We notice that the operator F + has the same form of the operator L 0 in (3.4.1) with

. The remainder R 0 satisfies the assumption (A2) by the discussion above. Notice also that, in view of (3.5.3), the constant given

(3.6.1)

Chapitre 3 -Reducibility of Schrödinger equation on a Zoll Manifold with unbounded potential

Moreover, if α ≤ β < 0 then

(3.6.11)

Proof. To prove (3.6.9) one reasons as in Lemma A.3 in [START_REF] Feola | Reducibility of Schrödinger equation on the sphere[END_REF]. The (3.6.11) and (3.6.10) follow by Lemma 3.6.1.

Lemma 3.6.5. One has

for any h ∈ s (see (3.2.15)) and β ∈ R.

Proof. One reasons as in Lemma A.4 in [START_REF] Feola | Reducibility of Schrödinger equation on the sphere[END_REF].

Flows of pseudo differential operators

Lemma 3.6.6. Fix m ≤ 0, 0 ≤ δ ≤ 1, r > d/2 and ρ ≥ 0 and consider S 1 ∈ A γ,O m,r and S 2 ∈ A γ,O δ,r (see Definition 3.2.3). Assume also that

where •, • is the standard L 2 scalar product. Let us define

For any s ≥ 0 there are ε 0 , C, p > 0 such that, for any

then the following holds true :

(3.6.25)

We have that sup

by (3.6.18) and the fact that r > d/2. Hence, using Duhamel formula and the (3.6.18), we deduce the (3.6.20) for k = 1. The (3.6.20) for k > 1 can be obtained in the same way by differentiating (3.6.25). The Lipschitz bounds (3.6.21)-(3.6.23) follows reasoning as in the estimates of ∂ ϕ Φ τ 2 (ϕ). The bounds (3.6.16), (3.6.17) can be deduced reasoning as done above and using the fact that the generator iS 1 (ϕ) is a bounded pseudodifferential operator.

Lemma 3.6.7. Let r 1 ≥ 0 and r > r

, where Φ τ 2 (ϕ) is given in Lemma 3.6.6. Then G 2 (ϕ) := Φ 2 (ϕ)R(ϕ)Φ -1 2 (ϕ) belongs to R γ,O ρ 1 ,r 1 . Moreover for any s ≥ 0 there exist p and C such that

(3.6.26)

Proof. We need to prove that the map ϕ → Γ(ϕ) is in H r 1 (T d ; L(H s ; H s+ρ 1 )). We note that

We estimate separately each summand in (3.6.27). First of all notice that, by the definition of the norm in (3.2.33) and the fact that r > r 1 + d/2, one has sup

(3.6.28)

Hence the summand in (3.6.27) with k 1 = k 3 = 0 is trivially bounded by the right hand side in (3.6.26). If at least one between k 1 , k 2 is different from zero we have, for any h ∈ H s , Similarly we prove in the bounded case :

. Moreover for any s ≥ 0 there exist p and C such that

(3.6.29) Data Availability. The data that supports the findings of this study are available within the article.

Denote I 0 = ( 4 3 π 2 -1, 4 9 π 4 ]. For any r ≥ 3, we define

Then our main result is stated as follow Theorem 4.2.1. Let r ≥ 1, 0 < µ 1. Then for any s ≥ s * , α > α * there exists a subset

and for any p ∈ I µ , for |t| ≤ µ -r+3/2 , there exists a transformation T : B s (µ/3) → B s (µ) satisfying

here Z is a polynomial of degree at most r + 2 that commutes with the actions N k , i.e.,

The canonical transformations and its inverse are close to identity

(4.2.10) Corollary 4.2.2. Fix r ≥ 1, assume Ω is nonresonant, then for any s sufficiently large, there exist µ 0 and c such that if the initial data is in

The difference between the "generic" system and the Kdv-Kdv system comes from D k . In the generic case b > 1 6 , we have |D k | ≤ 1, which makes it easy to prove tame property, while in the Kdv-Kdv case D k = 2πk is unbounded so that the nonlinear term P does not satisfies tame property. In the fullness of time, we would like to study this case, which is much more complicated. We also define the modulus of its vector field X f as

Definition 4.3.6. Let s ≥ 1, a homogeneous polynomial vector field X of degree r is said to be an s-tame map if there exists a constant C s such that

If a map is s-tame for any s ≥ 1 then it is said to be tame. Moreover if X is the vector field of a homogeneous polynomial f and X f is an s-tame map also, then we define r) 1 the s tame norm of f . We denote by T s M the set of function has finite s tame norm. Remark 4.3.7. If f ∈ T s M is a homogeneous polynomial of degree r + 1 then one has

The important property of s tame norm is that the Poisson bracket of two homogeneous polynomials in T s M is also in T s M (see lemma 4.12 in [START_REF] Bambusi | Birkhoff normal form for partial differential equations with tame modulus[END_REF]). Lemma 4.3.8. Assume that f, g ∈ T s M are two homogeneous polynomials of degrees n + 1 and m + 1 respectively, then {f, g} is a homogeneous polynomial of order n + m in

For a non homogeneous polynomial, we consider its Taylor expansion

where f r is homogeneous function of degree r. Then we define For the proof see [START_REF] Bambusi | Birkhoff normal form for partial differential equations with tame modulus[END_REF] proof of lemma 4.7.

Remind the idea of [BG06]

Here we recall the idea of the proof of the Theorem 2.13 in [START_REF] Bambusi | Birkhoff normal form for partial differential equations with tame modulus[END_REF]. The proof consists in using iterative Lie transformations. Consider an Hamiltonian function χ with the corresponding Hamiltonian equation

Denote by ϕ t χ the corresponding flow and call ϕ χ = ϕ 1 χ the Lie transform generated by χ. Then for any polynomial g, we have an important property

To prove the theorem we find Hamiltonian functions χ n and Hamiltonians H n such that

Here f n is a polynomial of order r, R n is a small term. This leads us to solve the homological equation

Proof of the Theorem 4.2.1

Since P has tame modulus, there exists a positive number µ # such that

here A = 3 2π|b| . Fix r * > 1, N > 1, for any r ≤ r * , we define µ r and µ * by

Then by induction, we can construct sequences of Hamiltonian χ r , H r , Z r and continuous functions f r for 0 < r ≤ r * as in (4.3.15). Precisely Proposition 4.5.1. Fix r * > 1, N > 1, for any r ≤ r * , there exists a canonical transformation ϕ r = ϕ χr which puts H in the form

here Z r is a polynomial of degree at most r + 2 having a zero of order 3 at the origin and is in (κ, α, N )-normal form ; f r is a polynomial having a zero of order r + 3 at the origin. Moreover

for any s ≥ 1. The transformation ϕ r satisfies sup (ψ,φ)∈Bs(µr)

(4.5.9)

Similarly, the inverse transformation fulfills the same estimate.

Proof of the Theorem 4.2.1

Proof. We proceed by induction. It is trivial for the case r = 0. For r > 0 we write Then by lemma 4.3.10, one has For the second term

Proof of theorem 4.2.1. We take T = ϕ χ 1 • • • • • ϕ χr , then Z = H 0 + Z r is in (κ, α, N ) normal form. We need to choose N in order to obtain R = R T r + R N r is small. We have

Choose N = N * = [µ -1 2rα ] and s > 2αr 2 + 1 then the last term is smaller than µ r+1 . The