  Flat band systems can yield interesting phenomena, such as dispersion
suppression of waves with frequency at the band. While linear transport
vanishes, the corresponding nonlinear case is still an open question. Here, we
study power transmission along nonlinear sawtooth lattices due to waves with
the flat band frequency injected at one end. While there is no power transfer
for small intensity, there is a threshold amplitude above which a surge of
power transmission occurs, i.e., supratransmission, for defocusing
nonlinearity. This is due to a nonlinear evanescent wave with the flat band
frequency that becomes unstable. We show that dispersion suppression and
supratransmission also exist even when the band is nearly flat.
