[Mesures gaussiennes quasi invariantes pour l'équation de Schrödinger non linéaire cubique avec dispersion d'ordre trois]
Dans cet article, nous considérons l'équation de Schrödinger non linéaire cubique avec dispersion d'ordre trois sur le cercle. Dans le cas non résonant, nous prouvons que les mesures gaussiennes de moyenne nulle sur les espaces de Sobolev , , sont quasi invariantes par le flot. En établissant le résultat, nous appliquons des transformations de gauge pour éliminer la partie résonante de la dynamique, et nous utilisons l'invariance des mesures gaussiennes par rapport à ces transformations de gauge.
In this paper, we consider the cubic nonlinear Schrödinger equation with third-order dispersion on the circle. In the non-resonant case, we prove that the mean-zero Gaussian measures on Sobolev spaces , , are quasi-invariant under the flow. In establishing the result, we apply gauge transformations to remove the resonant part of the dynamics and use invariance of the Gaussian measures under these gauge transformations.
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@article{CRMATH_2019__357_4_366_0,
author = {Oh, Tadahiro and Tsutsumi, Yoshio and Tzvetkov, Nikolay},
title = {Quasi-invariant {Gaussian} measures for the cubic nonlinear {Schr\"odinger} equation with third-order dispersion},
journal = {Comptes Rendus. Math\'ematique},
pages = {366--381},
publisher = {Elsevier},
volume = {357},
number = {4},
year = {2019},
doi = {10.1016/j.crma.2019.04.001},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2019.04.001/}
}
TY - JOUR AU - Oh, Tadahiro AU - Tsutsumi, Yoshio AU - Tzvetkov, Nikolay TI - Quasi-invariant Gaussian measures for the cubic nonlinear Schrödinger equation with third-order dispersion JO - Comptes Rendus. Mathématique PY - 2019 SP - 366 EP - 381 VL - 357 IS - 4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2019.04.001/ DO - 10.1016/j.crma.2019.04.001 LA - en ID - CRMATH_2019__357_4_366_0 ER -
%0 Journal Article %A Oh, Tadahiro %A Tsutsumi, Yoshio %A Tzvetkov, Nikolay %T Quasi-invariant Gaussian measures for the cubic nonlinear Schrödinger equation with third-order dispersion %J Comptes Rendus. Mathématique %D 2019 %P 366-381 %V 357 %N 4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2019.04.001/ %R 10.1016/j.crma.2019.04.001 %G en %F CRMATH_2019__357_4_366_0
Oh, Tadahiro; Tsutsumi, Yoshio; Tzvetkov, Nikolay. Quasi-invariant Gaussian measures for the cubic nonlinear Schrödinger equation with third-order dispersion. Comptes Rendus. Mathématique, Tome 357 (2019) no. 4, pp. 366-381. doi : 10.1016/j.crma.2019.04.001. http://www.numdam.org/articles/10.1016/j.crma.2019.04.001/
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