On a nonlocal analog of the Kuramoto-Sivashinsky equation

Rafael Granero-Belinchón, John K. Hunter

Published 2014-08-09, updated 2015-01-06Version 2

We study a nonlocal equation, analogous to the Kuramoto-Sivashinsky equation, in which short waves are stabilized by a possibly fractional diffusion of order less than or equal to two, and long waves are destabilized by a backward fractional diffusion of lower order. We prove the global existence, uniqueness, and analyticity of solutions of the nonlocal equation and the existence of a compact attractor. Numerical results show that the equation has chaotic solutions whose spatial structure consists of interacting traveling waves resembling viscous shock profiles.