An Abstract Birkhoff Normal Form Theorem and Exponential Type Stability of the 1d NLS

Abstract

We study stability times for a family of parameter dependent nonlinear Schrödinger equations on the circle, close to the origin. Imposing a suitable Diophantine condition (first introduced by Bourgain), we prove a rather flexible Birkhoff Normal Form theorem, which implies, e.g., exponential and sub-exponential time estimates in the Sobolev and Gevrey class respectively. Contents 1. Introduction and main results 1.1. Stability results 1.2. The abstract Birkhoff Normal Form Acknowledgements Part 1. An abstract framework for Birkhoff normal form on sequences spaces 2. Symplectic structure and Hamiltonian flows 3. Immersions for spaces of Hamiltonians. 4. Small divisors and homological equation 5. Abstract Birkhoff Normal Form Part 2. Applications to Gevrey and Sobolev cases 6. Immersions 7. Homological equation 8. Birkhoff normal form 9. Gevrey stability. Proof of Theorem 1.1 10. Sobolev stability Part 3. Appendices Appendix A. Constants. Appendix B. Proofs of the main properties of the norms Appendix C. Small divisor estimates References 1 Namely g is a holomorphic function on the domain Ta := {x ∈ C/2πZ : |Im x| < a} with L 2 -trace on the boundary. 2 A vector ω ∈ R n is called diophantine when it is badly approximated by rationals, i.e. it satisfies, for some γ, τ > 0, |k • ω| ≥ γ|k| −τ , ∀k ∈ Z n \ {0} .

About linearization of infinite-dimensional Hamiltonian systems

“…A similar result was proved in [CMW]. An interesting feature of the last three papers is that instead on relying on tameness properties they use the fact that the equations they study have some symmetries, namely they are gauge and translation invariant (actually in [BMP18] the translation invariance condition is weakened). All the preceding results regard semilinear PDEs.…”

“…Successively Faou and Grébert in [FG13] considered the case of analytic initial data and proved subexponential bounds on the stability time for classes of NLS equations in T d . In [BMP18] the first author with Biasco and Massetti studied an abstract Birkhoff normal form on sequence spaces proving subexponential stability times for Gevrey regular initial data. A similar result was proved in [CMW].…”

“…Hence we endow Ω with the product topology and with the corresponding probability measure. With respect to such measure Diophantine frequencies are typical, namely Ω \ D γ has measure proportionally bounded by γ (see [BMP18][Lemma 4.1]). Then we prove :…”

Almost-Periodic Response Solutions for a Forced Quasi-Linear Airy Equation

“…Then (5.3), (5.9), (5.10) guarantee that the hypotheses of Lemmata A.10-A.11 are verified. Hence, we apply Lemma A.10-(ii) to expand the operator e −G Pe G − P, Lemma A.11-(ii) to expand e −G ∂ 3 x e G − ∂ 3…”

“…More recently there have been results such as [20,26,27], which use a KAM approach. We mention also [3,4,11,29] which however are tailored for an autonomous PDE.…”