  The modulational instability of spatially uniform states in the nonlinear
Schr\"odinger equation is examined in the presence of higher-order dissipation.
The study is motivated by results on the effects of three-body recombination in
Bose-Einstein condensates, as well as by the important recent work of Segur et
al. on the effects of linear damping in NLS settings. We show how the presence
of even the weakest possible dissipation suppresses the instability on a longer
time scale. However, on a shorter scale, the instability growth may take place,
and a corresponding generalization of the MI criterion is developed. The
analytical results are corroborated by numerical simulations. The method is
valid for any power-law dissipation form, including the constant dissipation as
a special case.
