On a nonlocal analog of the Kuramoto-Sivashinsky equation

TL;DR

This paper investigates a nonlocal version of the Kuramoto-Sivashinsky equation, proving fundamental properties of solutions and revealing chaotic behavior with traveling wave structures through numerical simulations.

Contribution

It introduces a novel nonlocal equation with fractional diffusion terms and establishes existence, uniqueness, and analyticity of solutions, along with the existence of a compact attractor.

Findings

Existence and uniqueness of solutions confirmed.

Solutions exhibit analyticity and form a compact attractor.

Numerical simulations reveal chaotic solutions with traveling wave structures.

Abstract

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.