Mathematics

We first discuss local in time well-posedness for a large class of fully nonlinear Schrödinger equation  on the circle. Furthermore, under suitable algebraic  hypotheses on the nonlinearity, we prove  the long time existence and stability of solutions with small, and even in x, initial condition.Both the results are achieved by proving a priori energy estimates on the Sobolev norms of the solutions. The key ingredients of the proof are:

1. the use of para-differential calculus techniques to reduce the equation to a constant coefficients ones up to smoothing remainders; 
2. the construction  of modified energy for the solution by means of Birkhoff normal forms techniques.