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.
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].
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