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 (∂/∂t)P(x,t)=D(∂^γ/∂x^γ)[P(x,t)]^ν. Exact time-dependent solutions are found for ν=(2-γ)/(1+γ)(-∞<γ<~2). By considering the long-distance asymptotic behavior of these solutions, a connection is established, namely, q=(γ+3)/(γ+1)(0<γ<~2), with the solutions optimizing the nonextensive entropy characterized by index q. Interestingly enough, this relation coincides with the one already known for L\'evy-like superdiffusion (i.e., ν=1 and 0<γ<~2). Finally, for (γ,ν)=(2,0) we obtain $q=5/3,$ which differs from the value $q=2$ corresponding to the γ=2 solutions available in the literature (ν<1 porous medium equation), thus exhibiting nonuniform convergence.