We review recent results on the nonlinear Schrödinger equation
where and . In any space dimension and for any , we construct a class of (arbitrarily large) initial values for which there exists a local solution. Moreover, if , we construct a class of (arbitrarily large) initial values for which there exists a global solution that scatters as . If and , we construct a class of (arbitrarily large) initial values for which there exists a global solution, of which we give a precise asymptotic expansion as (of modified scattering type). These results rely on the construction of solutions that do not vanish, so as to avoid any issue related to the lack of regularity of the nonlinearity at . To study the asymptotic behavior, we apply the pseudo-conformal transformation. This yields the desired asymptotic behavior if . In the case , a further step is required, and we estimate the solutions by allowing a certain growth of the Sobolev norms, which depends on the order of regularity through a cascade of exponents.
@article{SLSEDP_2017-2018____A1_0, author = {Thierry Cazenave and Ivan Naumkin}, title = {Spatial behavior for {NLS} and applications to scattering}, journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications}, note = {talk:1}, pages = {1--11}, publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {2017-2018}, doi = {10.5802/slsedp.116}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.116/} }
TY - JOUR AU - Thierry Cazenave AU - Ivan Naumkin TI - Spatial behavior for NLS and applications to scattering JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:1 PY - 2017-2018 SP - 1 EP - 11 PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique UR - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.116/ DO - 10.5802/slsedp.116 LA - en ID - SLSEDP_2017-2018____A1_0 ER -
%0 Journal Article %A Thierry Cazenave %A Ivan Naumkin %T Spatial behavior for NLS and applications to scattering %J Séminaire Laurent Schwartz — EDP et applications %Z talk:1 %D 2017-2018 %P 1-11 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.116/ %R 10.5802/slsedp.116 %G en %F SLSEDP_2017-2018____A1_0
Thierry Cazenave; Ivan Naumkin. Spatial behavior for NLS and applications to scattering. Séminaire Laurent Schwartz — EDP et applications (2017-2018), Talk no. 1, 11 p. doi : 10.5802/slsedp.116. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.116/
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