Anomalous scaling and solitary waves in systems with nonlinear diffusion

Abstract

We study a nonlinear convective-diffusive equation, local in space and time, which has its background in the dynamics of the thickness of a wetting film. The presence of a nonlinear diffusion predicts the existence of fronts as well as shock fronts. Despite the absence of memory effects, solutions in the case of pure nonlinear diffusion exhibit an anomalous subdiffusive scaling. Due to a balance between nonlinear diffusion and convection we, in particular, show that solitary waves appear. For large times they merge into a single solitary wave exhibiting a topological stability. Even though our results concern a specific equation, numerical simulations support the view that anomalous diffusion and the solitary waves disclosed will be general features in such nonlinear convective-diffusive dynamics.

Hyperballistic Superdiffusion and Explosive Solutions to the Non-Linear Diffusion Equation

“…The diffusion of grains in granular media considered by Christov and Stone [25] is yet another example. Pritchard et al, [26] studied gravity-driven fluid flow in layered porous media finding that the fluid motion could be described by a concentrationdependent diffusivity as did Hansen et al [27] for the spreading of wetting films in wedges. Anomalous diffusion in random geometries, fractals and tree-like structures has been studied for decades [28][29][30][31][32].…”

Anomalous Diffusion in Systems with Concentration-Dependent Diffusivity: Exact Solutions and Particle Simulations

“…Common to all of these examples is that γ is negative, ranging from c −5 [4] to c −1/2 [14]. That is, in all these examples the diffusivity increases with increasing concentration.…”

“…The diffusion of grains in granular media considered by Christov and Stone [12] is yet another example. Pritchard et al [13] studied gravity-driven fluid flow in layered porous media finding that the fluid motion could be described by a concentration-dependent diffusivity as did Hansen et al [14] for the spreading of wetting films in wedges.…”

Anomalous Diffusion in Systems with Concentration-Dependent Diffusivity

“…One finds a dependence of the diffusivity on concentration in many physical systems. Newman considers examples from population dynamics and combustion [4], Azevedo et al study water ingress in zeolites [5,6], Fischer et al [7] and Christov and Stone [8] consider diffusion of grains in granular media, Hansen et al [9] the dynamics of wetting films in wedges. Anomalous diffusion is reported in all of these papers.…”

“…In the papers that assume the diffusivity to take the form (3) [4,[7][8][9], the exponent γ is assumed to be negative.…”