  We investigate Beliaev-Landau scattering in a gas of interacting photons in a
coherently driven array of nonlinear dissipative resonators, as described by
the 1D driven-dissipative Bose-Hubbard model. Due to the absence of detailed
balance in such an out-of-equilibrium setup, steady-state properties can be
much more sensitive to the underlying microscopic dynamics. As the popular
truncated Wigner approximation dramatically fails in capturing this physics, we
present an alternative approach, based on a systematic expansion beyond the
Bogoliubov approximation, which includes the third-order correlation functions
in the dynamics. As experimentally accessible signatures of Beliaev-Landau
processes, we report a small but nonnegligible correction to the Bogoliubov
prediction for the steady-state momentum distribution showing a characteristic
series of peaks and dips, as well as non-Gaussian features in the statistics of
the cavity output field.
