Backreaction of axion coherent oscillations
  We investigate how coherent oscillations backreact on the evolution of the
condensate wave function of ultra-light axions in the non-relativistic regime
appropriate to cosmic structure formation. The coherent oscillations induce
higher harmonics beyond the fundamental mode considered so far when a
self-interaction is present, and imprint oscillations in the gravitational
potential. We emphasize that the effective self-interaction felt by the
slowly-varying envelop of the wave function always differs from the bare
Lagrangian interaction potential. We also point out that, in the hydrodynamical
formulation of the Gross-Pitaevskii equation, oscillations in the gravitational
potential result in an attractive force that counteracts the effect of the
quantum pressure arising from the strong delocalization of the particles. Since
these effects become significant on physical scales less than the (large)
Compton wavelength of the particle, they are presumably not very relevant on
the mildly nonlinear scales traced by intergalactic neutral hydrogen for axion
masses consistent with the bounds from the Lyman-$\alpha$ forest. However, they
might affect the formation of virialized cosmological structures and their
stability.
