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

TL;DR

This paper proves that fully nonlinear, reversible, parity-preserving Schrödinger equations on the circle have solutions that exist for very long times when starting from small, even initial data, by using paralinearization and Birkhoff normal forms.

Contribution

It establishes long time existence for a broad class of fully nonlinear Schrödinger equations on the circle under small initial data and non-resonance conditions, using advanced para-differential and normal form techniques.

Findings

Solutions exist for time scales of order ^{-N} for any N with small initial data.

Diagonalization of the system up to regularizing terms.

Construction of modified energies via Birkhoff normal forms.

Abstract

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.