Mathematical models of generalized diffusion

We discuss in this paper equations describing processes involving non-linea r and higher-order diffusion. We focus on a particular case (u(t) = 2 lambd a (2)(uu(x))(x) + lambda (2)u(xxxx)), which is put into analogy with the Kd V equation. A balance of nonlinearity and higher-order diffusion enables th e existence of self-similar solutions, describing diffusive shocks. These s hocks are continuous solutions with a discontinuous higher-order derivative at the shock front. We argue that they play a role analogous to the solito n solutions in the dispersive case. We also discuss several physical instan ces where such equations are relevant.