Interaction of cavity solitons on an unstable background

The interaction of two cavity solitons in a driven semiconductor laser above lasing threshold is investigated. We focus on the case in which the background field of the solitons is turbulent because the laser is below the injection locking point. We show that the solitons move spontaneously and either reach some equilibrium distance or merge. Different behaviors are found depending on how far from the injection locking point the laser is. The laser is modeled by a set of effective Maxwell-Bloch equations which include an equation for the macroscopic polarization that mimics the complex susceptibility of the semiconductor. In that way we avoid the emergence of an unphysical behavior of the background which instead appears when the polarization is adiabatically eliminated, which amounts to assuming infinite gain linewidth. The simulations are slow because the time scales of the different dynamical variables differ by four orders of magnitude. Yet, we show that the results of the complete set of equations can be accurately reproduced with a reduced set of equations where the polarization is adiabatically eliminated but a diffusion term is included in Maxwell equation, which accounts for the finiteness of the gain linewidth.