  Solitons, which describe the propagation of concentrated beams of light
through nonlinear media, can exhibit a variety of behaviors as a result of the
intrinsic dissipation, diffraction, and the nonlinear effects. One of these
phenomena, modeled by the complex Ginzburg-Landau equation, are chaotic
explosions, transient enlargements of the soliton that may induce random
transversal displacements, which in the long run lead to a random walk of the
soliton center. As we show in this work, the transition from non-moving to
moving solitons is not a simple bifurcation but includes a sequence of normal
and anomalous random walks. We analyze their statistics with the distribution
of generalized diffusivities, a novel approach that has been used successfully
for characterizing anomalous diffusion.
