  In this article we develop an effective theory of pulse propagation in a
nonlinear and disordered medium. The theory is formulated in terms of a
nonlinear diffusion equation. Despite its apparent simplicity this equation
describes novel phenomena which we refer to as "locked explosion" and
"diffusive" collapse. In this sense the equation can serve as a paradigmatic
model, that can be applied to such distinct physical systems as laser beams
propagating in disordered photonic crystals or Bose-Einstein condensates
expanding in a disordered environment.
