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

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 non-quadratic 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 Schr¨odinger system studied by Métivier, Rauch, Texier, Zumbrun, Ekeland, Séré. Using a “free flow component” decomposition and applying an abstract Nash-Moser-H¨ormander theorem, we improve the existing results regarding both the size of the data and the regularity of the solutions.