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We analyze the spread of a localized peak of energy into vacuum for nonlineardiffusive processes. In contrast with standard diffusion, the nonlinearityresults in a compact wave with a sharp front separating the perturbed regionfrom vacuum. In $d$ spatial dimensions, the front advances as $t^{1/(2+da)}$according to hydrodynamics, with $a$ the nonlinearity exponent. We show thatfluctuations in the front position grow as $\sim t^{\mu} \eta$, where$\mu<1/(2+da)$ is a new exponent that we measure and $\eta$ is a randomvariable whose distribution we characterize. Fluctuating corrections tohydrodynamic profiles give rise to an excess penetration into vacuum, revealingscaling behaviors and robust features. We also examine the discharge of anonlinear rarefaction wave into vacuum. Our results suggest the existence ofuniversal scaling behaviors at the fluctuating level in nonlinear diffusion.

Compact Waves in Microscopic Nonlinear Diffusion