Resonant vector bundles, conjugate points, and the stability of pulse solutions to the {S}wift-{H}ohenberg equation using validated numerics: Part I

TL;DR

This paper introduces a new theoretical framework for using validated numerics to rigorously analyze the stability of pulse solutions in the Swift-Hohenberg equation, especially addressing challenges posed by resonances in vector bundles.

Contribution

It extends existing methods by developing a theory for resonant vector bundles, enabling the detection of conjugate points and eigenvalues even in resonant cases.

Findings

Developed a new approach to handle resonances in vector bundles.

Provided a method to detect conjugate points using validated numerics.

Laid groundwork for rigorous stability analysis of PDE solutions.

Abstract

In this paper, we develop new theory connected with resonant vector bundles that will allow for the use of validated numerics to rigorously determine the stability of pulse solutions in the context of the Swift-Hohenberg equation. For many PDEs, the stability of stationary solutions is determined by the absence of point spectra in the open right half of the complex plane. Recently, theoretical developments have allowed one to use objects called conjugate points to detect such unstable eigenvalues for certain linearized operators. Moreover, in certain cases these conjugate points can themselves be detected using validated numerics. The aim of this work is to extend this framework to contexts where the vector bundles, which control the existence of conjugate points, have certain resonances. Such resonances can prevent the use of standard (though involved) techniques in computer assisted proofs, and in this paper we provide a method to overcome this obstacle. Due to its length, the analysis has been divided into two parts: Part I in the present work, and Part II in [BJPS25].