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.

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

References in corpus (4)

• The interaction of Kelvin waves and the non-locality of the energy transfer in superfluids
• Gravity capillary standing water waves
• Local well-posedness for quasi-linear NLS with large Cauchy data on the circle
• Quasi invariant modified Sobolev norms for semi linear reversible PDEs

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• Long time solutions for quasi-linear Hamiltonian perturbations of Schrödinger and Klein-Gordon equations on tori
• One dimensional energy cascades in a fractional quasilinear NLS
• Exponential stability of solutions to the Schr{ö}dinger-Poisson equation
• Dynamics of quintic nonlinear Schr{ö}dinger equations in $H^{2/5+}(\mathbb{T})$