Long time existence for fully nonlinear NLS with small Cauchy data on the circle

Abstract

In this paper we prove long time existence for a large class of fully nonlinear, reversible and parity preserving Schrödinger equations on the one dimensional torus. We show that for any initial condition even in $x$, regular enough and of size $\epsilon$ sufficiently small, the lifespan of the solution is of order ${\epsilon }^{-N}$ for any $N\in \mathbb{N}$ if some non resonance conditions are fulfilled. After a paralinearization of the equation we perform several para-differential changes of variables which diagonalize the system up to a very regularizing term. Once achieved the diagonalization, we construct modified energies for the solution by means of Birkhoff normal forms techniques.

Some typos have been corrected, the bibliography has been corrected. Accepted paper on "Annali della Scuola Normale Superiore di Pisa"