Geodesic
Size of data in implicit function problems and singular perturbations for nonlinear Schrödinger systems
Annales de l'I.H.P. Analyse non linéaire, Tome 40 (2023) no. 5, pp. 1051-1091

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We investigate a general question about the size and regularity of the data and the solutions in implicit function problems with loss of regularity. First, we give a heuristic explanation of the fact that the optimal data size found by Ekeland and Séré with their recent nonquadratic version of the Nash–Moser theorem can also be recovered, for a large class of nonlinear problems, with quadratic schemes. Then we prove that this heuristic observation applies to the singular perturbation Cauchy problem for the nonlinear Schrödinger system studied by Métivier, Rauch, Texier, Zumbrun, Ekeland, and Séré. Using a “free flow component” decomposition and applying an abstract Nash–Moser–Hörmander theorem, we improve the existing results regarding both the size of the data and the regularity of the solutions.

Accepté le :
Publié le :
DOI : 10.4171/aihpc/56
Classification : 47J07, 35B25, 35Q55, 35L52
Keywords: Nash–Moser theorem, nonlinear Schrödinger equation, nonlinear Schrödinger system, Cauchy problem, singular perturbation, size of data 
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     title = {Size of data in implicit function problems and singular perturbations for nonlinear {Schr\"odinger} systems},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1051--1091},
     volume = {40},
     number = {5},
     year = {2023},
     doi = {10.4171/aihpc/56},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/aihpc/56/}
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Baldi, Pietro; Haus, Emanuele. Size of data in implicit function problems and singular perturbations for nonlinear Schrödinger systems. Annales de l'I.H.P. Analyse non linéaire, Tome 40 (2023) no. 5, pp. 1051-1091. doi: 10.4171/aihpc/56

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