In this paper we consider a class of fully nonlinear forced and reversible Schrödinger equations and prove existence and stability of quasi-periodic solutions. We use a Nash-Moser algorithm together with a reducibility theorem on the linearized operator in a neighborhood of zero. Due to the presence of the highest order derivatives in the non-linearity the classic KAM-reducibility argument fails and one needs to use a wider class of changes of variables such has diffeomorphisms of the torus and pseudo-differential operators. This procedure automtically produces a change of variables, well defined on the phase space of the equation, which diagonalizes the operator linearized at the solution. This gives the linear stability.

We prove reducibility of a class of quasi-periodically forced linear equations of the form, a is a small, C ∞ function, Q is a pseudo differential operator of order −1, provided that ω ∈ R ν satisfies appropriate non-resonance conditions. Such PDEs arise by linearizing the Degasperis-Procesi (DP) equation at a small amplitude quasi-periodic function. Our work provides a first fundamental step in developing a KAM theory for perturbations of the DP equation on the circle. Following [3], our approach is based on two main points: first a reduction in orders based on an Egorov type theorem then a KAM diagonalization scheme.In both steps the key difficulites arise from the asymptotically linear dispersion law. In view of the application to the nonlinear context we prove sharp tame bounds on the diagonalizing change of variables. We remark that the strategy and the techniques proposed are applicable for proving reducibility of more general classes of linear pseudo differential first order operators. * This research was supported by PRIN 2015 "Variational methods, with applications to problems in mathematical physics and geometry" and by ERC grant "Hamiltonian PDEs and small divisor problems: a dynamical systems approach n. 306414 under FP7".

In this paper we prove reducibility of classes of linear first order operators on tori by applying a generalization of Moser's theorem on straightening of vector fields on a torus. We consider vector fields which are a C ∞ perturbations of a constant vector field, and prove that they are conjugated -by a C ∞ torus diffeomorphism-to a constant diophantine flow, provided that the perturbation is small in some given H s1 norm and that the initial frequency is in some Cantorlike set. Actually in the classical results of this type the regularity of the change of coordinates which straightens the perturbed vector field coincides with the class of regularity in which the perturbation is required to be small. This improvement is achieved thanks to ideas and techniques coming from the Nash-Moser theory.

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In this paper we prove long time existence for a large class of fully nonlinear, reversible and parity preserving Schrödinger equations on the one dimensional torus. We show that for any initial condition even in x, regular enough and of size ε sufficiently small, the lifespan of the solution is of order ε −N for any N ∈ N if some non resonance conditions are fulfilled. After a paralinearization of the equation we perform several para-differential changes of variables which diagonalize the system up to a very regularizing term. Once achieved the diagonalization, we construct modified energies for the solution by means of Birkhoff normal forms techniques. * This research was supported by PRIN 2015 "Variational methods, with applications to problems in mathematical physics and geometry".The following is a subclass of the previous class made of those operators which are autonomous, i.e. they depend on the variable t only through the function U .Definition 2.5 (Autonomous smoothing operator). We define, according to the notation of Definition 2.2, the class of autonomous non-homogeneous smoothing operator R −ρ K,0,N [r, aut] as the subspace of R −ρ K,0,N [r] made of those maps (U, V ) → R(U )V satisfying estimates (2.8) with K ′ = 0, the time dependence being only through U = U (t). In the same way, we denote by ΣR −ρ K,0,p [r, N, aut] the space of maps (U, V ) → R(U, V ) of the form (2.9) with K ′ = 0 and where the last term belongs to R −ρ K,0,N [r, aut].0,N [r, aut]. This inclusion follows by the multi-linearity of R in each argument, and by estimate (2.6). For further details we refer to the remark after Definition 2.2.3 in [7]. Spaces of MapsIn the following, sometimes, we shall treat operators without having to keep track of the number of lost derivatives in a very precise way. We introduce some further classes.. The following is a subclass of the class defined in 2.16 made of those symbols which depend on the variable t only through the function U .Definition 2.18 (Autonomous non-homogeneous Symbols). We denote by Γ m K,0,p [r, aut] the subspace of Γ m K,0,p [r] made of the non-homogeneous symbols (U, x, ξ) → a(U ; x, ξ) that satisfy estimate (2.18) with K ′ = 0, the time dependence being only through U = U (t).Remark 2.19. A symbol a(U; ·) of Γ m p defines, by restriction to the diagonal, the symbol a(U, . . . , U ; ·) for Γ m K,0,p [r, aut] for any r > 0. For further details we refer the reader to the first remark after Definition 2.1.3 in [7].

We prove local in time well-posedness for a large class of quasilinear Hamiltonian, or parity preserving, Schrödinger equations on the circle. After a paralinearization of the equation, we perform several paradifferential changes of coordinates in order to transform the system into a paradifferential one with symbols which, at the positive order, are constant and purely imaginary. This allows to obtain a priori energy estimates on the Sobolev norms of the solutions. * This research was supported by PRIN 2015 "Variational methods, with applications to problems in mathematical physics and geometry".

We discuss a Nash-Moser/ KAM algorithm for the construction of invariant tori for tame vector fields. Similar algorithms have been studied widely both in finite and infinite dimensional contexts: we are particularly interested in the second case where tameness properties of the vector fields become very important. We focus on the formal aspects of the algorithm and particularly on the minimal hypotheses needed for convergence. We discuss various applications where we show how our algorithm allows to reduce to solving only linear forced equations. We remark that our algorithm works at the same time in analytic and Sobolev class.

We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth, and prove a rigorous reduction of these equations to Bikhoff normal form up to degree four. This proves a conjecture of Zakharov-Dyachenko [62] based on the formal Birkhoff integrability of the water waves Hamiltonian truncated at order four. As a consequence, we also obtain a long-time stability result: periodic perturbations of a flat interface that are of size ε in a sufficiently smooth Sobolev space lead to solutions that remain regular and small up to times of order ε −3 .Main difficulties in the proof are the quasilinear nature of the equations, the presence of small divisors arising from near-resonances, and non-trivial resonant four-waves interactions, the so-called Benjamin-Feir resonances. The main ingredients that we use are: (1) various reductions to constant coefficient operators through flow conjugation techniques; (2) the verification of key algebraic properties of the gravity water waves system which imply the integrability of the equations at nonnegative orders; (3) smoothing procedures and Poincaré-Birkhoff normal form transformations; (4) a normal form identification argument that allows us to handle Benajamin-Feir resonances by comparing with the formal computations of [62,22,30,20].