The stability of nonlinear Schrödinger equations with a potential in high Sobolev norms revisited

Abstract

We consider the nonlinear Schrödinger equations with a potential on $\mathbb T^d$. For almost all potentials, we show the almost global stability in very high Sobolev norms. We apply an iteration of the Birkhoff normal form, as in the formulation introduced by Bourgain [4]. This result reproves a dynamical consequence of the infinite dimensional Birkhoff normal form theorem by Bambusi and Grebert [2].