We review results from two recent articles [34, 35] on the asymptotic completeness of small standing solitary waves for a class of one-dimensional nonlinear Schrödinger equations. The models considered are perturbations of the integrable cubic 1D Schrödinger equation. The notion of internal modes plays an important role and part of the discussion concerns their existence. In case an internal mode exists, the proof of asymptotic stability is more delicate and relies on a nonlinear variant of the Fermi golden rule.
@article{SLSEDP_2023-2024____A7_0, author = {Yvan Martel}, title = {Asymptotic stability of small solitons for one-dimensional~nonlinear {Schr\"odinger} equations}, journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications}, note = {talk:10}, pages = {1--22}, publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {2023-2024}, doi = {10.5802/slsedp.170}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.170/} }
TY - JOUR AU - Yvan Martel TI - Asymptotic stability of small solitons for one-dimensional nonlinear Schrödinger equations JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:10 PY - 2023-2024 SP - 1 EP - 22 PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique UR - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.170/ DO - 10.5802/slsedp.170 LA - en ID - SLSEDP_2023-2024____A7_0 ER -
%0 Journal Article %A Yvan Martel %T Asymptotic stability of small solitons for one-dimensional nonlinear Schrödinger equations %J Séminaire Laurent Schwartz — EDP et applications %Z talk:10 %D 2023-2024 %P 1-22 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.170/ %R 10.5802/slsedp.170 %G en %F SLSEDP_2023-2024____A7_0
Yvan Martel. Asymptotic stability of small solitons for one-dimensional nonlinear Schrödinger equations. Séminaire Laurent Schwartz — EDP et applications (2023-2024), Talk no. 10, 22 p. doi : 10.5802/slsedp.170. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.170/
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