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

Massimiliano Berti; Philippe Bolle

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 1, page 229-286
- ISSN: 1435-9855

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topBerti, 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 -

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