# Traveling wave oscillatory patterns in a signed Kuramoto-Sivashinsky equation with absorption

## TL;DR

This paper proves the existence and stability of a periodic traveling wave pattern in a signed Kuramoto-Sivashinsky equation with absorption, using a computer-assisted mathematical approach.

## Contribution

It provides a partial proof of a conjecture on the existence and stability of periodic orbits in a complex fourth-order differential equation related to wave patterns.

## Key findings

- Existence of a periodic orbit confirmed.
- Orbit is asymptotically stable.
- Computer-assisted proof methodology established.

## Abstract

In this paper, a partial proof of a conjecture raised by Galaktionov and Svirshchevskii concerning existence and global uniqueness of an asymptotically stable periodic orbit in a fourth-order piecewise linear ordinary differential equation is presented. The fourth-order equation comes from the study of traveling wave patterns in a signed Kuramoto-Sivashinsky equation with absorption. The proof is twofold. First, the problem of solving for the periodic orbit is transformed into a zero finding problem in four dimensional space, which is solved with a computer-assisted proof based on Newton's method and the contraction mapping theorem. Second, the rigorous bounds about the periodic orbit in phase space are combined with the theory of discontinuous dynamical systems to prove that the orbit is asymptotically stable.