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 Schrödinger equations on the one dimensional torus. We show that, if some non-resonance conditions are fulfilled, for any N ∊ ℕ and for any initial condition, which is even in x and size ε in an appropriate Sobolev space, the lifespan of the solution is of order ε-N. 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.