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

TL;DR

This paper investigates a non-linear convective-diffusive system related to wetting film dynamics, revealing anomalous sub-diffusive scaling and the emergence of stable solitary waves through a balance of diffusion and convection.

Contribution

It demonstrates the existence of anomalous scaling and solitary waves in a specific non-linear convective-diffusive equation, supported by numerical simulations suggesting generality.

Findings

Solutions exhibit anomalous sub-diffusive scaling.

Solitary waves form and merge into a stable single wave.

Numerical results support the generality of these phenomena.

Abstract

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.