Size of data in implicit function problems and singular perturbations for nonlinear Schrödinger systems

Abstract

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.