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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 wellas shock fronts. Despite the absence of memory effects, solutions in the caseof pure non-linear diffusion exhibit an anomalous sub-diffusive scaling. Due toa balance between non-linear diffusion and convection we, in particular, showthat solitary waves appear. For large times they merge into a single solitarywave exhibiting a topological stability. Even though our results concern aspecific equation, numerical simulations supports the view that anomalousdiffusion and the solitary waves disclosed will be general features in suchnon-linear convective-diffusive dynamics.

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