Synchronization and Competition in a Double-Bump-on-Tail Instability

We experimentally and theoretically consider a double-bump-on-tail instability by mapping the general wave-kinetic problem to a multiple beam propagation problem using statistical light. More specifically, we consider the nonlinear interaction of three spatially-incoherent beams in a self-focusing photorefractive crystal. For weak nonlinearity, we observe instability competition and sequential flattening of the bumps in momentum space, with no observable variations in position-space intensity. This joint dynamics resembles the phase synchronization of a ``classical'' laser system (relaxation from a ``non- equilibrium'' state to a lower-energy one), with the corresponding gain rates following from the optical equivalent of inverse Landau damping. For strong nonlinearity, intensity modulations appear and the triple-hump spectrum merges into a single-peaked profile with an algebraic $k^{-2}$ inertial range. This spectrum, with its associated modulations, is a definitive observation of soliton, or Langmuir, turbulence.