The relation between the size of perturbations and the dimension of tori in an infinite-dimensional KAM theorem of Pöschel

Li, Xuemei; Liu, Shujuan

doi:10.1016/j.na.2020.111754

Cited by 3 publications

(1 citation statement)

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“…We could take this comparison further and assume that all the actions I j with j ∈ S are equidistributed. Then of course we would see a dependence between the size of the perturbation and the dimension m as in [1] or in [35]. Note that even if these constants are surely non optimal, in any case the size of the perturbation shrinks to zero more than factorially with the dimension of the torus.…”

Section: Biasco Je Massetti and M Procesimentioning

confidence: 98%

Almost periodic invariant tori for the NLS on the circle

Biasco

Massetti

Procesi

2021

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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In this paper we study the existence and linear stability of almost periodic solutions for a NLS equation on the circle with external parameters. Starting from the seminal result of Bourgain in [15] on the quintic NLS, we propose a novel approach allowing to prove in a unified framework the persistence of finite and infinite dimensional invariant tori, which are the support of the desired solutions. The persistence result is given through a rather abstract "counter-term theorem" à la Herman, directly in the original elliptic variables without passing to action-angle ones. Our framework allows us to find "many more" almost periodic solutions with respect to the existing literature and consider also non-translation invariant PDEs.

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Section: Biasco Je Massetti and M Procesimentioning

confidence: 98%