Anomalous diffusion associated with nonlinear fractional derivative Fokker-Planck-like equation: Exact time-dependent solutions

We consider the $d=1$ nonlinear Fokker-Planck-like equation with fractional derivatives $frac{partial}{partial t}P(x,t)=D frac{partial^{gamma}}{partial x^{gamma}}[P(x,t) ]^{ u}$. Exact time-dependent solutions are found for $ u = frac{2-gamma}{1+ gamma}$ ($-infty<gamma leq 2$). By considering the long-distance {it asymptotic} behavior of these solutions, a connection is established, namely $q=frac{gamma+3}{gamma+1}$ ($0<gamma le 2$), with the solutions optimizing the nonextensive entropy characterized by index $q$ . Interestingly enough, this relation coincides with the one already known for Levy-like superdiffusion (i.e., $ u=1$ and $0<gamma le 2$). Finally, for $(gamma, u)=(2, 0)$ we obtain $q=5/3$ which differs from the value $q=2$ corresponding to the $gamma=2$ solutions available in the literature ($ u<1$ porous medium equation), thus exhibiting nonuniform convergence.