A NEKHOROSHEV TYPE THEOREM FOR THE FRACTIONAL NONLINEAR SCHRÖDINGER EQUATION

We prove a Nekhoroshev type theorem for the fractional nonlinear Schrödinger equation under Dirichlet boundary conditions. More precisely, our findings show that the solutions with $ \varepsilon $-small initial data in the Gevrey space remain in their small magnitude over time intervals of order $ \varepsilon^{-\lvert \ln\varepsilon\rvert^{\gamma}} $ with $ 0 <\gamma<1/10 $. The result can be proved by using Birkhoff normal form method and the so-called tame property of the nonlinearity in Gevrey space.