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  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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Jessica Elisa Massetti, Michela Procesi, Luca Biasco, Almost periodic invariant tori for the NLS on the circle. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 3, pp. 711–758DOI 10.1016/J.ANIHPC.2020.09.003