D. E. Pelinovsky
Traveling waves in fractional models
Fractional Dispersive Models and Applications: Recent Developments and Future Perspectives,
Editors: P. H Kevrekidis, J. Cuevas-Maraver; Nonlinear Systems and Complexity 37 (Springer Nature, Switzerland, 2024) 155-186
Abstract:
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.
Keywords:
Fractional KdV equations; existence of traveling periodic waves; variational characterization; linear stability;