Journal of the European Mathematical Society


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Volume 15, Issue 1, 2013, pp. 229–286
DOI: 10.4171/JEMS/361

Quasi-periodic solutions with Sobolev regularity of NLS on $\mathbb T^d$ with a multiplicative potential

Massimiliano Berti[1] and Philippe Bolle[2]

(1) Dipartimento di Matematica e Applicazioni, Università degli Studi di Napoli Federico II, Via Cintia, Monte S. Angelo 4, 80126, NAPOLI, ITALY
(2) Laboratoire d'Analyse non Linéaire et Géométrie (E, Université d'Avignon et des Pays de Vaucluse, 84018, AVIGNON, FRANCE

We prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on $\mathbb T^d, d \geq 1$, finitely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are $C^{\infty}$ then the solutions are $C^{\infty}$. The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized operators ("Green functions") along scales of Sobolev spaces. The key off-diagonal decay estimates of the Green functions are proved via a new multiscale inductive analysis. The main novelty concerns the measure and "complexity" estimates.

Keywords: Nonlinear Schrödinger equation, Nash–Moser theory, KAM for PDE, quasi-periodic solutions, small divisors, in

Berti M, Bolle P. Quasi-periodic solutions with Sobolev regularity of NLS on $\mathbb T^d$ with a multiplicative potential. J. Eur. Math. Soc. 15 (2013), 229-286. doi: 10.4171/JEMS/361