  We consider turbulence of waves that interact weakly via four-wave scattering
(sea waves, plasma waves, spin waves, and many others). In the first
non-vanishing order in the interaction, the occupation number of waves satisfy
a closed kinetic equation which has stationary solutions describing turbulent
cascades. We show that a straightforward perturbation theory beyond the kinetic
equation gives terms that generally diverge both at small (IR) and large (UV)
wavenumbers for a direct cascade. The analysis up to the third order identifies
the most UV-divergent terms. In order to gain qualitative analytic control, we
sum a subset of the most UV divergent term, to all orders, giving a
perturbation theory which is generally free from UV divergence, showing that
turbulence becomes independent of the dissipation scale when it goes to zero.
On the contrary, the ever-present IR divergence means that the effective
coupling is parametrically larger than the naive estimate and grows with the
pumping scale (similar to anomalous scaling in fluid turbulence). This suggests
that the kinetic equation does not describe wave turbulence even of arbitrarily
small level if the cascade is sufficiently long. We show that the character of
strong turbulence is determined by the sign of the coupling, that is, whether
the effective four-wave interaction is enhanced or suppressed by collective
effects. The enhancement possibly signals that strong turbulence is dominated
by multi-wave bound states (solitons, shocks, cusps), similar to confinement in
quantum chromodynamics.
