I discuss recent results that show that
spatial homogeneity, the symmetry property that all statistical moments
are functions
only of the relative geometry of any configuration of points,
can be spontaneously broken by an instability
of the finite flux Kolmogorov-Zakharov spectrum in certain
(usually one dimensional) systems. As a result,
wave turbulence is not a statistical attractor in this case.
As I show for a particular system (the
one-dimensional Majda-McLaughlin-Tabak equation), the new nonequilibrium
state is dominated by radiating solitary pulses. The evolution
of these pulses becomes the main mechanism of transfer of energy
from long spatial scales to short scales. The spectrum of this
system (the Majda-McLaughlin-Tabak spectrum) can be derived
from the evolution of these pulses.