On the well-posedness of the periodic fractional Schrödinger equation

We consider the periodic fractional nonlinear Schrödinger equation $$i{u}_t-(-\mathrm{\Delta }{)}^{\frac{s}{2}}u+\mathcal{N}(\mathrm{\mid }u\mathrm{\mid })u=0,x\in {\mathbb{T}}^N,t\in \mathbb{R},s>0,$$ where the nonlinearity term is expressed in two ways: the first one $\mathcal{N}\in C^J({\mathbb{R}}^{+})$, whose derivatives have a certain polynomial decay, e.g., $\mathcal{N}(\mathrm{\mid }u\mathrm{\mid })=\log (\mathrm{\mid }u\mathrm{\mid })$; the second one is given by a sum of powers, possibly infinite, $$\mathcal{N}(\mathrm{\mid }u\mathrm{\mid })=\sum a_k\mathrm{\mid }u{\mathrm{\mid }}^{{\gamma }_k},{\gamma }_k\in \mathbb{R},a_k\in \mathbb{C},$$ which includes examples such as $\mathcal{N}(\mathrm{\mid }u\mathrm{\mid })u=\frac{u}{\mathrm{\mid }u{\mathrm{\mid }}^{\gamma }},$ $\gamma >0$. By using standard properties of periodic Sobolev spaces $H^J({\mathbb{T}}^N)$, $J>0$, we study the local well-posedness for the Cauchy problems of the above equations when initial data satisfy a non-vanishing condition $\underset{x\in {\mathbb{T}}^N}{\inf }\mathrm{\mid }{u}_0(x)\mathrm{\mid }>0$.