Davey-Stewartson system (DS) is given by a couple of equations that model the time evolution of $2D$ surface of water waves that propagate predominantly in one direction but the wave amplitude is modulated slowly in the horizontal directions. DS systems are very well studied in terms of well-posedness and stability, blow-up profiles, existence of standing and travelling waves. We will essentially discuss smoothing properties of DS in the initial value problem setting. To reach the result we will introduce some preliminaries including Tao’s $[k; Z]$ multiplier method.