  It is well known that wave collapses can emerge from the focusing
one-dimensional (1-D) Majda-McLaughlin-Tabak (MMT) model as a result of
modulational instability. However, how these wave collapses affect the spectral
properties and statistics of the wave field has not been adequately studied. We
undertake this task by simulating the forced-dissipated 1-D MMT model over a
range of forcing amplitudes. Our results show that when the forcing is weak,
the spectrum agrees well with the prediction by wave turbulence theory with few
collapses in the field. As the forcing strength increases, we see an increase
in the occurrence of collapses, together with a transition from a power-law
spectrum to an exponentially decaying spectrum. Through a spectral
decomposition, we find that the exponential spectrum is dominated by the wave
collapse component in the non-integrable MMT model, which is in analogy to a
soliton gas in integrable turbulence.
