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

In this paper we prove long time existence for a large class of fully nonlinear, reversible and parity preserving Schr\"odinger equations on the one dimensional torus. We show that for any initial condition even in $x$, regular enough and of size $\varepsilon$ sufficiently small, the lifespan of the solution is of order $\varepsilon^{-N}$ 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.