Anomalous Scaling and Solitary Waves in Systems with Non-Linear Diffusion

We study a non-linear 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 non-linear diffusion predicts the existence of fronts as well as shock fronts. Despite the absence of memory effects, solutions in the case of pure non-linear diffusion exhibit an anomalous sub-diffusive scaling. Due to a balance between non-linear 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 supports the view that anomalous diffusion and the solitary waves disclosed will be general features in such non-linear convective-diffusive dynamics.Comment: Corrected typos, added 3 references and 2 figure

Anomalous diffusion on clusters in steady-state two-phase flow in porous media in two dimensions

We study diffusion processes on clusters of non-wetting fluid dispersed in a wetting one flowing in a two-dimensional porous medium under steady-state conditions using a numerical model. At the critical saturation and capillary number of 10-5, where the cluster size distribution follows a power law, we find anomalous diffusion characterized by two critical exponents, $d_{{\rm rw}\parallel }$ = 2.35 ± 0.05 in the average flow direction and $d_{{\rm rw}\perp }$ = 3.51 ± 0.05 in the perpendicular direction. We determine the conductivity exponents to be $\mu _{\parallel }$ = 1.6 ± 0.2 and $\mu _{\perp }$ = 1.25 ± 0.1, respectively. The high-frequency scaling exponents of the AC conductivity we find to be $\eta _{\parallel }$ =0. 37 ± 0.1 and $\eta _{\perp }$ = 0.36 ± 0.1, respectively