  In this report, we consider a semi-infinite discrete nonlinear Schr\"odinger
equation driven at one edge by a driving force. The equation models the
dynamics of coupled waveguide arrays. When the frequency of the forcing is in
the allowed-band of the system, there will be a linear transmission of energy
through the lattice. Yet, if the frequency is in the upper forbidden band, then
there is a critical driving amplitude for a nonlinear tunneling, which is
called supratransmission, of energy to occur. Here, we discuss mathematically
the mechanism and the source of the supratransmission. By analyzing the
existence and the stability of the rapidly decaying static discrete solitons of
the system, we show rigorously that two of the static solitons emerge and
disappear in a saddle-node bifurcation at a critical driving amplitude. One of
the emerging solitons is always stable in its existence region and the other is
always unstable. We argue that the critical amplitude for supratransmission is
then the same as the critical driving amplitude of the saddle-node bifurcation.
We consider as well the case of the forcing frequency in the lower forbidden
band. It is discussed briefly that there is no supratransmission because in
this case there is only one rapidly decaying static soliton that exists and is
stable for any driving amplitude.
