Geodesic
Quasi-periodic solutions with Sobolev regularity of NLS on $\mathbb T^d$ with a multiplicative potential
Journal of the European Mathematical Society, Tome 15 (2013) no. 1, pp. 229-286 Cet article a éte moissonné depuis la source EMS Press

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We prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on Td,d≥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∞ then the solutions are C∞. 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.
DOI : 10.4171/jems/361
Classification : 35-XX, 37-XX, 00-XX
Keywords: Nonlinear Schrödinger equation, Nash–Moser theory, KAM for PDE, quasi-periodic solutions, small divisors, in 
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     author = {Massimiliano Berti and Philippe Bolle},
     title = {Quasi-periodic solutions with {Sobolev} regularity of {NLS} on $\mathbb T^d$ with a multiplicative potential},
     journal = {Journal of the European Mathematical Society},
     pages = {229--286},
     year = {2013},
     volume = {15},
     number = {1},
     doi = {10.4171/jems/361},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/361/}
}
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Massimiliano Berti; Philippe Bolle. Quasi-periodic solutions with Sobolev regularity of NLS on $\mathbb T^d$ with a multiplicative potential. Journal of the European Mathematical Society, Tome 15 (2013) no. 1, pp. 229-286. doi: 10.4171/jems/361

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