2011
DOI: 10.1103/physreve.83.056314
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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 … Show more

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Cited by 3 publications
(5 citation statements)
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“…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].…”
Section: Introductionmentioning
confidence: 95%
“…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].…”
Section: Introductionmentioning
confidence: 95%
“…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.…”
Section: Introductionmentioning
confidence: 96%
“…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.…”
Section: Introductionmentioning
confidence: 97%
“…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.…”
Section: Introductionmentioning
confidence: 99%
“…In the papers that assume the diffusivity to take the form (3) [4,[7][8][9], the exponent γ is assumed to be negative.…”
Section: Introductionmentioning
confidence: 99%