Effects of dissipative disorder on front formation in pattern forming systems

We study the effects of weak disorder in the linear gain coefficient on front formation in pattern forming systems described by the cubic-quintic nonlinear Schrodinger equation. We calculate the statistics of the front amplitude and position. We show that the distribution of the front amplitude has a loglognormal diverging form at the maximum possible amplitude and that the distribution of the front position has a lognormal tail. The theory is in good agreement with our numerical simulations. We show that these results are valid for other types of dissipative disorder and relate the loglognormal divergence of the amplitude distribution to the form of the emerging front tail.