The goal of this monograph is to prove that any solution of the Cauchy problem for the
capillary-gravity water waves equations, in one space dimension, with periodic, even in …

We study stability times for a family of parameter dependent nonlinear Schrödinger
equations on the circle, close to the origin. Imposing a suitable Diophantine condition (first …

In this paper we prove a result of almost global existence for some abstract nonlinear PDEs
on flat tori and apply it to some concrete equations, namely a nonlinear Schrödinger …

We prove an abstract implicit function theorem with parameters for smooth operators defined
on scales of sequence spaces, modeled for the search of quasi-periodic solutions of PDEs …

We prove an almost global existence result for space periodic solutions of the 1D gravity-
capillary water waves equations with constant vorticity. The result holds for any value of …

We prove an extended lifespan result for the full gravity-capillary water waves system with a
2 dimensional periodic interface: for initial data of sufficiently small size ε ε, smooth solutions …

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 …

We consider general classes of nonlinear Schrödinger equations on the circle with nontrivial
cubic part and without external parameters. We construct a new type of normal forms …

We consider a one-parameter family of beam equations with Hamiltonian non-linearity in
one space dimension under periodic boundary conditions. In a unified functional framework …

The initial value problem for the cubic defocusing nonlinear Schr\" odinger equation $
i\partial_t u+\Delta u=| u|^ 2 u $ on the plane is shown to be globally well-posed for initial …