Let us consider the fractional Schrödinger equation with *periodic boundary conditions*

$$

begin{cases}

iu_tmathbf{+}(-Delta)^{alpha}u= pm |u|^2u,; x in mathbb{T}, t in mathbb{R}_+\

u(x,0)=u_0(x), ; x in mathbb{T} tag{1}

end{cases}

$$

where $mathbb{T} subset mathbb{R}$ is a torus, $u_0 in H^s(mathbb{T})$, $alpha in (1/2,1)$ and $s>0$. Also, in the same way, consider

$$

begin{cases}

iu_tmathbf{-}(-Delta)^{alpha}u= pm |u|^2u,; x in mathbb{T}, t in mathbb{R}_+\

u(x,0)=u_0(x), ; x in mathbb{T}. tag{2}

end{cases}

$$

**Question**. What is difference between the equation $(1)$ and $(2)$? My question is the following sense: in $(1)$ the authors showed (see Theorem $2$) that the $(1)$ is globally well-posed in $H^s(mathbb{T})$ provided that $s> tfrac{10 alpha+1}{12}.$

In this way, is true the same result for the equation $(2)$? If no, why? It’s not clear to me what kind of properties the equation loses (or gains) by changing the sign. Note that, we are in a periodic context.

References are welcome.

$(1)$ S. Demirbas, M. B. Erdogan, and N. Tzirakis. *Existence and uniqueness theory for the fractional Schrödinger equation on the torus*. In Some topics in harmonic analysis and applications, volume $34$ of Adv. Lect. Math.(ALM), pages $145$–$162$. Int. Press, Somerville, MA, $2016$.