Long time stability of fractional nonlinear Schrödinger equations

Abstract

We investigate the long time stability of the solutions to the fractional nonlinear Schrödinger (FNLS) equation under periodic boundary condition$i{\psi }_t+{(-\Delta )}^{s_0}\psi +\frac{\partial F\left(\right|\psi {|}^2)}{\partial \bar{\psi }}=0,x\in T,t\in R,s_0\in (3/4,1),$ where ${(-\Delta )}^{s_0}$ denotes the Riesz fractional differentiation defined in [18]. Here $F\left(z\right)$ is a real-valued polynomial function of z, fulfilling $F^{\prime }\left(z\right){|}_{z=0}=0$, $F^{″}\left(z\right){|}_{z=0}\ne 0$. Our findings indicate that for all $s_0\in (3/4,1)$ and almost all R-small initial data in Sobolev norm, the corresponding solutions remain their small magnitude over time-intervals of length $R^{-|\ln R{|}^{\gamma }}$ with $0<R\ll 1$, $0<\gamma <1/5$.