We consider a nonlinear Schrödinger system with fractional diffusion
$$\left\{\begin{array}{cc}{\left(\Delta \right)}^{\alpha \u22152}u\left(x\right)+A\left(x\right)u\left(x\right)={v}^{p}\left(x\right)\phantom{\rule{1em}{0ex}}\hfill & \text{in}\Omega ,\hfill \\ {\left(\Delta \right)}^{\beta \u22152}v\left(x\right)+B\left(x\right)v\left(x\right)={u}^{q}\left(x\right)\phantom{\rule{1em}{0ex}}\hfill & \text{in}\Omega ,\hfill \\ u\left(x\right)\ge 0,v\left(x\right)\ge 0\phantom{\rule{1em}{0ex}}\hfill & \text{in}\Omega ,\hfill \\ u\left(x\right)=v\left(x\right)=0\phantom{\rule{1em}{0ex}}\hfill & \text{on}{\Omega}^{C},\hfill \end{array}\right.$$ 
where
$\Omega $
is an unbounded parabolic domain. We first establish a narrow region principle.
Using this principle and a direct method of moving planes, we obtain the
monotonicity of nonnegative solutions and the Liouvilletype result for the nonlinear
Schrödinger system with fractional diffusion. We also obtain the radially
symmetric result of positive solutions for the system in the unit ball when
$A\left(x\right)$ and
$B\left(x\right)$ are
constants.
