  Periodically-driven quantum systems make it possible to reach stationary
states with new emerging properties. However, this process is notoriously
difficult in the presence of interactions because continuous energy exchanges
generally boil the system to an infinite temperature featureless state. Here,
we describe how to reach nontrivial states in a periodically-kicked
Gross-Pitaevskii disordered system. One ingredient is crucial: both disorder
and kick strengths should be weak enough to induce sufficiently narrow and
well-separated Floquet bands. In this case, inter-band heating processes are
strongly suppressed and the system can reach an exponentially long-lived
prethermal plateau described by the Rayleigh-Jeans distribution. Saliently, the
system can even undergo a wave condensation process when its initial state has
a sufficiently low total quasi-energy. These predictions could be tested in
nonlinear optical experiments or with ultracold atoms.
