Traveling Waves in Fractional Models

Fractional models of the Korteweg-de Vries (KdV) type are discussed in the context of propagation of one-dimensional traveling waves in nonlocal nonlinear dispersive systems. Spatially periodic waves can be constructed by using small-amplitude expansions, fixed-point methods, and calculus of variations. The existence theory is closely related to the stability theory, both of which provide the first step towards understanding of the nonlinear dynamics of traveling periodic waves in such nonlocal systems. Recent existence and stability results on the traveling periodic waves are reviewed for the fractional KdV models with quadratic and cubic nonlinearities.