Size of data in implicit function problems and singular perturbations for nonlinear Schr\"odinger systems

TL;DR

This paper explores the size and regularity of data and solutions in implicit function problems, demonstrating that quadratic schemes can recover optimal data size and applying this to improve results in nonlinear Schrödinger systems.

Contribution

It shows that quadratic schemes can achieve optimal data size in implicit problems and applies this insight to enhance results for nonlinear Schrödinger systems.

Findings

Quadratic schemes recover optimal data size.

Application to nonlinear Schrödinger systems.

Improved regularity and data size results.

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 S\'er\'e 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 M\'etivier, Rauch, Texier, Zumbrun, Ekeland, S\'er\'e. 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.