  The integrable Davey-Stewartson system is a linear combination of the two
elementary flows that commute: $\mathrm{i} q_{t_1} + q_{xx} +
2q\partial_y^{-1}\partial_x (|q|^2) =0$ and $\mathrm{i} q_{t_2} + q_{yy} +
2q\partial_x^{-1}\partial_y (|q|^2) =0$. In the literature, each elementary
Davey-Stewartson flow is often called the Fokas system because it was studied
by Fokas in the early 1990s. In fact, the integrability of the Davey-Stewartson
system dates back to the work of Ablowitz and Haberman in 1975; the elementary
Davey-Stewartson flows, as well as another integrable $(2+1)$-dimensional
nonlinear Schr\"odinger equation $\mathrm{i} q_{t} + q_{xy} + 2
q\partial_y^{-1}\partial_x (|q|^2) =0$ proposed by Calogero and Degasperis in
1976, appeared explicitly in Zakharov's article published in 1980. By applying
a linear change of the independent variables, an elementary Davey-Stewartson
flow can be identified with a $(2+1)$-dimensional generalization of the
integrable long wave-short wave interaction model, called the Yajima-Oikawa
system: $\mathrm{i} q_{t} + q_{xx} + u q=0$, $u_t + c u_y = 2(|q|^2)_x$. In
this paper, we propose a new integrable semi-discretization (discretization of
one of the two spatial variables, say $x$) of the Davey-Stewartson system by
constructing its Lax-pair representation; the two elementary flows in the
semi-discrete case indeed commute. By applying a linear change of the
continuous independent variables to an elementary flow, we also obtain an
integrable semi-discretization of the $(2+1)$-dimensional Yajima-Oikawa system.
