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

Margaret Beck, Jonathan Jaquette, Hannah Pieper

TL;DR

The paper develops a rigorous framework for stability analysis of pulse solutions to the Swift–Hohenberg equation by integrating conjugate-point theory with validated numerics in the presence of vector-bundle resonances. It extends the parameterization method to construct high-order, convergent representations of invariant manifolds and their resonant vector bundles, and provides computable bounds L^ ext{±}_{conj} to confine conjugate points to a compact domain. The approach yields a robust, computer-assisted pathway to certify the stability of pulses via a single-λ eigenvalue analysis (λ=0) and a posteriori verification of discarding conjugate points, with Part II to instantiate a full stability computation for a specific pulse. Together, these contributions advance rigorous stability proofs for pattern-forming PDEs by handling universal bundle resonances and enabling large-scale validated computations. The work has implications for reliable stability assessments in Hamiltonian PDEs and broadens the scope of computer-assisted proofs in infinite-dimensional dynamics.

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].

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

The paper develops a rigorous framework for stability analysis of pulse solutions to the Swift–Hohenberg equation by integrating conjugate-point theory with validated numerics in the presence of vector-bundle resonances. It extends the parameterization method to construct high-order, convergent representations of invariant manifolds and their resonant vector bundles, and provides computable bounds L^ ext{±}_{conj} to confine conjugate points to a compact domain. The approach yields a robust, computer-assisted pathway to certify the stability of pulses via a single-λ eigenvalue analysis (λ=0) and a posteriori verification of discarding conjugate points, with Part II to instantiate a full stability computation for a specific pulse. Together, these contributions advance rigorous stability proofs for pattern-forming PDEs by handling universal bundle resonances and enabling large-scale validated computations. The work has implications for reliable stability assessments in Hamiltonian PDEs and broadens the scope of computer-assisted proofs in infinite-dimensional dynamics.

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].

Paper Structure

This paper contains 29 sections, 24 theorems, 306 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background: analytic parameterizations of manifolds & bundles
  3. Multi-index Notation
  4. Parameterization of an invariant manifold
  5. Parameterization of bundles in the non-resonant case
  6. Parameterization of bundles in the resonant case
  7. Conjugacy Equation.
  8. Choice of normal form.
  9. Homological equation.
  10. Example with vector bundle resonance: the bistable equation
  11. Parameterization of the stable manifold
  12. Parameterization of the stable/unstable bundles
  13. Closed forms of the stable/unstable bundles
  14. Proving the conjugate points lie in a compact domain
  15. Determination of $L^-_{\mathrm{conj}}$
  16. ...and 14 more sections

Key Result

Lemma 2.3

CabreFontichdelaLlave03aCabreFontichdelaLlave03bCabreFontichdelaLlave05 Let $P: B^m_\delta(0) \subset \mathbb{R}^m \to \mathbb{R}^n$ be a smooth function. Then for all $\sigma \in B^m_\delta(0)$, $P$ satisfies if and only if $P$ satisfies

Figures (6)

  • Figure 1: Schematic depiction of the path of $\mathbb{E}^{\mathrm{u}}_-(x; 0)$. The plane $\ell_\mathrm{sand}$ is depicted as the shaded rectangle. The dotted orange line, light blue dot, green dot, purple trajectory, and yellow star all form various components of our computer assisted proof.
  • Figure 2: Flowchart indicating which parts of the computer assisted proof are contained in the present paper, Part I (red), and which are contained in Part II (blue), BeckJaquettePieperStorm25.
  • Figure 3: The conjugacy described by Equation \ref{['E:invar']}.
  • Figure 4: Expanding ($\mu > 0$) and contracting ($\mu < 0$) vector bundles.
  • Figure 5: Plot of the stable manifold (blue), the analytic bundle (green) and the invariant bundle (orange).
  • ...and 1 more figures

Theorems & Definitions (62)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7: vandenBergJames16
  • Definition 2.8
  • Lemma 2.9: vandenBergJames16
  • Definition 2.10
  • ...and 52 more