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.
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.
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Tadahiro Oh 1; Yoshio Tsutsumi 2; Nikolay Tzvetkov 3
@article{CRMATH_2019__357_4_366_0, author = {Tadahiro Oh and Yoshio Tsutsumi and Nikolay Tzvetkov}, 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}, }
TY - JOUR AU - Tadahiro Oh AU - Yoshio Tsutsumi AU - Nikolay Tzvetkov 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 DO - 10.1016/j.crma.2019.04.001 LA - en ID - CRMATH_2019__357_4_366_0 ER -
%0 Journal Article %A Tadahiro Oh %A Yoshio Tsutsumi %A Nikolay Tzvetkov %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 %R 10.1016/j.crma.2019.04.001 %G en %F CRMATH_2019__357_4_366_0
Tadahiro Oh; Yoshio Tsutsumi; Nikolay Tzvetkov. Quasi-invariant Gaussian measures for the cubic nonlinear Schrödinger equation with third-order dispersion. Comptes Rendus. Mathématique, Volume 357 (2019) no. 4, pp. 366-381. doi : 10.1016/j.crma.2019.04.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.04.001/
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