Rogue wave light bullets of the three-dimensional inhomogeneous nonlinear Schr?dinger equation

We discover single and homocentric optical spheres of the three-dimensional inhomogeneous nonlinear Schr?dinger equation(NLSE) with spherical symmetry, which is a novel model of light bullets that can present a three-dimensional rogue wave. The isosurface of this light bullet oscillates along the radius direction and does not travel with the evolution of time. The localized nature of rogue wave light bullets both in space and in time, which is in complete contrast to the traveling character of the usual light bullets, is due to the localization of the rogue wave in the one-dimensional NLSE. We present also an investigation of the stability of the optical sphere solutions. The lower modes of perturbation are found to display transverse instabilities that break the spherical symmetry of the system. For the higher modes, the optical sphere solutions can be classified as stable solutions.