  Recently, the existence of robust three-dimensional light bullets (LBs) was
predicted theoretically in the output of a laser coupled to a distant saturable
absorber. In this manuscript, we analyze the stability and the range of
existence of these dissipative localized structures and provide guidelines and
realistic parameter sets for their experimental observation. In order to reduce
the complexity of the analysis, we first approximate the three-dimensional
problem by a reduced equation governing the dynamics of the transverse profile.
This effective theory provides an intuitive picture of the LB formation
mechanism. Moreover, it allows us to perform a detailed multi-parameter
bifurcation study and to identify the different mechanisms of instability. It
is found that the LBs experience dominantly either homogeneous oscillation or
symmetry breaking transversal waves radiation. In addition, our analysis
reveals several non-intuitive scaling behaviors as functions of the linewidth
enhancement factors and the saturation parameters. Our results are confirmed by
direct numerical simulations of the full system.
