Anomalous scaling and solitary waves in systems with nonlinear diffusion

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.