Fractional Integrable Nonlinear Soliton Equations

Nonlinear integrable equations serve as a foundation for nonlinear dynamics, and fractional equations are well known in anomalous diffusion. We connect these two fields by presenting the discovery of a new class of integrable fractional nonlinear evolution equations describing dispersive transport in fractional media. These equations can be constructed from nonlinear integrable equations using a widely generalizable mathematical process utilizing completeness relations, dispersion relations, and inverse scattering transform techniques. As examples, this general method is used to characterize fractional extensions to two physically relevant, pervasive integrable nonlinear equations: the Korteweg–deVries and nonlinear Schrödinger equations. These equations are shown to predict superdispersive transport of nondissipative solitons in fractional media.

Figure 1

Localized waves predicted by the fKdV and fNLS equations (22)–(24) show superdispersive transport as their velocity increases as $\epsilon$ increases from 0 to 1. Like anomalous diffusion where the mean squared displacement is proportional to $t^{\alpha }$, the velocity in anomalous dispersion is proportional to $A^{\epsilon }$, where $A$ is the amplitude of the wave. The parameter values used are $\kappa =3/2$, $\xi =2$, and $\eta =1/2$.

Figure 2

Note that soliton solutions to the fKdV equation propagate without dissipating or spreading out. The parameter values used are $\kappa =3/2$ and $x_0=0$.