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 (partial derivative/partial derivative t)P(x,t) =D(partial derivative(gamma)/partial derivative x(gamma))[P(x,t)](nu). Exact time-dependent solutions are found for nu=(2- gamma)/(1 + gamma)(-infinity< y less than or equal to 2). By considering the long-distance asymptotic behavior of these solutions, a connection is established, namely, q =(gamma+ 3)/(y + 1)(0<gamma less than or equal to 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., nu = 1 and 0<gamma less than or equal to 2). Finally, for (gamma,nu)=(2,0) we obtain q=5/3, which differs from the value q=2 corresponding to the gamma=2 solutions available in the literature (nu<1 porous medium equation), thus exhibiting nonuniform convergence.