Voir la notice de l'article provenant de la source Numdam
We consider the cubic Schrödinger (NLS) equation on the Euclidean space perturbed by a short-range potential . The presence of a negative simple eigenvalue for gives rise to a curve of small and localized nonlinear ground states that yield some time-periodic solutions known to be asymptotically stable in the energy space. We study the persistence of these coherent states under rough perturbations. We shall construct a large measure set of small scaling-supercritical solutions below the energy space that display some asymptotic stability behaviors. The main difficulty is the need to handle the interactions of localized and dispersive terms in the modulation equations. To do so, we use a critical-weighted strategy to combine probabilistic nonlinear estimates in critical spaces based on , (controlling higher-order terms) with some local energy decay estimates (controlling lower-order terms). We also revisit in the perturbed setting the analysis of Bényi, Oh, and Pocovnicu (2015) on the probabilistic global well-posedness and scattering for small supercritical initial data. We use a distorted Fourier transform and semiclassical functional calculus to generalize probabilistic and bilinear Strichartz estimates.
@article{AIHPC_2022__39_6_1261_0,
author = {Camps, Nicolas},
title = {Asymptotic stability of small ground states for {NLS} under random perturbations},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1261--1318},
volume = {39},
number = {6},
year = {2022},
doi = {10.4171/aihpc/31},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4171/aihpc/31/}
}
TY - JOUR AU - Camps, Nicolas TI - Asymptotic stability of small ground states for NLS under random perturbations JO - Annales de l'I.H.P. Analyse non linéaire PY - 2022 SP - 1261 EP - 1318 VL - 39 IS - 6 UR - http://geodesic.mathdoc.fr/articles/10.4171/aihpc/31/ DO - 10.4171/aihpc/31 LA - en ID - AIHPC_2022__39_6_1261_0 ER -
%0 Journal Article %A Camps, Nicolas %T Asymptotic stability of small ground states for NLS under random perturbations %J Annales de l'I.H.P. Analyse non linéaire %D 2022 %P 1261-1318 %V 39 %N 6 %U http://geodesic.mathdoc.fr/articles/10.4171/aihpc/31/ %R 10.4171/aihpc/31 %G en %F AIHPC_2022__39_6_1261_0
Camps, Nicolas. Asymptotic stability of small ground states for NLS under random perturbations. Annales de l'I.H.P. Analyse non linéaire, Tome 39 (2022) no. 6, pp. 1261-1318. doi: 10.4171/aihpc/31
Cité par Sources :