Fractional Integrable Dispersive Equations

Fractional equations arise widely in physical systems and nonlinear integrable systems are fundamentally important in the study of nonlinear dynamics. This chapter discusses the recently discovered methodology of Ablowitz, Been, Carr (ABC) that shows how these two fields can be connected. The ABC construction employs three key parts: the dispersion relation of the linearized problem, completeness of underlying eigenfunctions and the inverse scattering transform. The equations discussed in this paper are prototypical; they include the fractional integrable: Korteweg-deVries (KdV), nonlinear Schrödinger (NLS), modified KdV, sine/sinh-Gordon and discrete NLS equations. The fractional systems obtained are nonlocal. When the fractional parameter vanishes the nonlocality disappears and the equations reduce to their well-known local counterparts. Applications include nonlinear wave propagation in multiscale media ranging from the naturally occurring bumpy, wrinkled, corrugated, rough, and crumpled geometries of nature to artificially constructed lattices with selected Fourier components beyond simple ordered lattices but stopping short of full disorder.