The Cauchy problem for the fractional nonlinear Schrödinger equation in Sobolev space

Abstract

We investigate the Cauchy problem for the fractional nonlinear Schrödinger (fNLS) equation. First, we establish the local well-posedness in the fractional Sobolev spaces $H^{\gamma}$ with $\gamma\geq 0$ for the fNLS equation under less regularity assumptions for the nonlinear term than previous work. Based on the local well-posedness result, we then prove that the fNLS equation is globally well-posed in $H^{\gamma}$ with $\gamma\geq 0$ if $4\sigma/d \leq \nu \leq \nu_c (\gamma )$ and the initial data is sufficiently small.