JournalsjemsVol. 15, No. 1pp. 229–286

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

  • Massimiliano Berti

    Università degli Studi di Napoli Federico II, Italy
  • Philippe Bolle

    Université d'Avignon et des Pays de Vaucluse, France
Quasi-periodic solutions with Sobolev regularity of NLS on $\mathbb T^d$ with a multiplicative potential cover
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Abstract

We prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on Td,d1\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 CC^{\infty} then the solutions are CC^{\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.

Cite this article

Massimiliano Berti, Philippe Bolle, Quasi-periodic solutions with Sobolev regularity of NLS on Td\mathbb T^d with a multiplicative potential. J. Eur. Math. Soc. 15 (2013), no. 1, pp. 229–286

DOI 10.4171/JEMS/361