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

Berti, Massimiliano, and Bolle, Philippe. "Quasi-periodic solutions with Sobolev regularity of NLS on $\mathbb {T}^d$ with a multiplicative potential." Journal of the European Mathematical Society 015.1 (2013): 229-286. <http://eudml.org/doc/277509>.

@article{Berti2013,
abstract = {We prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on $\mathbb \{T\}^d, d\ge 1$, 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.},
author = {Berti, Massimiliano, Bolle, Philippe},
journal = {Journal of the European Mathematical Society},
keywords = {nonlinear Schrödinger equation; Nash–Moser theory; KAM for PDE; quasi-periodic solutions; small divisors; infinite-dimensional Hamiltonian systems; nonlinear Schrödinger equation; Nash-Moser theory; KAM for PDE; quasi-periodic solutions; small divisors; infinite-dimensional Hamiltonian systems},
language = {eng},
number = {1},
pages = {229-286},
publisher = {European Mathematical Society Publishing House},
title = {Quasi-periodic solutions with Sobolev regularity of NLS on $\mathbb \{T\}^d$ with a multiplicative potential},
url = {http://eudml.org/doc/277509},
volume = {015},
year = {2013},
}

TY - JOUR
AU - Berti, Massimiliano
AU - Bolle, Philippe
TI - Quasi-periodic solutions with Sobolev regularity of NLS on $\mathbb {T}^d$ with a multiplicative potential
JO - Journal of the European Mathematical Society
PY - 2013
PB - European Mathematical Society Publishing House
VL - 015
IS - 1
SP - 229
EP - 286
AB - We prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on $\mathbb {T}^d, d\ge 1$, 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.
LA - eng
KW - nonlinear Schrödinger equation; Nash–Moser theory; KAM for PDE; quasi-periodic solutions; small divisors; infinite-dimensional Hamiltonian systems; nonlinear Schrödinger equation; Nash-Moser theory; KAM for PDE; quasi-periodic solutions; small divisors; infinite-dimensional Hamiltonian systems
UR - http://eudml.org/doc/277509
ER -