Compact Waves in Microscopic Nonlinear Diffusion

TL;DR

This paper investigates the propagation and fluctuations of compact waves in nonlinear diffusion processes, revealing universal scaling behaviors and characterizing front fluctuations and penetration into vacuum.

Contribution

It introduces a detailed analysis of fluctuating front positions and penetration depths in nonlinear diffusion, providing new exponents and characterizations of their distributions.

Findings

Front advances as t^{1/(2+da)} in d dimensions.

Fluctuations grow as t^{ta} with ta<1/(2+da).

Discharge of nonlinear rarefaction waves into vacuum exhibits universal scaling.

Abstract

We analyze the spread of a localized peak of energy into vacuum for nonlinear diffusive processes. In contrast with standard diffusion, the nonlinearity results in a compact wave with a sharp front separating the perturbed region from vacuum. In $d$ spatial dimensions, the front advances as $t^{1/(2+da)}$ according to hydrodynamics, with $a$ the nonlinearity exponent. We show that fluctuations 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 random variable whose distribution we characterize. Fluctuating corrections to hydrodynamic profiles give rise to an excess penetration into vacuum, revealing scaling behaviors and robust features. We also examine the discharge of a nonlinear rarefaction wave into vacuum. Our results suggest the existence of universal scaling behaviors at the fluctuating level in nonlinear diffusion.