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Some applications of fractional equations. (English) Zbl 1041.35073

The authors deal with the application of fractional equations in physics. They consider the kinetic equation with fractional derivatives

P(x,t) t ' = 2 P x '2 +ε α P |x ' | α ,1<α<2,(1)

where ε is a constant, and α P |x ' | α is the Riesz derivatives. The authors study the competition between normal diffusion and diffusion induced by fractional derivatives for (1). It is shown that for large times the fractional derivative term dominates the solution and leads to power type tails. Moreover a corresponding fractional generalization of the Ginzburg-Landau and nonlinear Schrödinger equations is proposed.

MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
26A33Fractional derivatives and integrals (real functions)