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Stability analysis for localized solutions in PDEs and nonlocal equations on $\mathbb{R}^m$

Matthieu Cadiot

TL;DR

This work develops a computer-assisted framework for linear stability analysis of localized solutions to PDEs and nonlocal equations on $\mathbb{R}^m$, explicitly separating the essential spectrum, given by the symbol range $\{l(\xi): \xi\in\mathbb{R}^m\}$, from discrete eigenvalues. By translating the problem to Fourier coefficients on bounded cubes and employing a pseudo-diagonalization with a generalized Gershgorin theorem, the authors obtain explicit, computable disks that enclose eigenvalues of the Jacobian at the true localized solution $\tilde{u}$, with controlled radii and multiplicities. A two-tier homotopy argument transfers spectral enclosures from a finite-dimensional, periodic surrogate $DF(U_0)$ to the full operator $D\mathbb{F}(\tilde{u})$, providing rigorous localization away from the essential spectrum. The methodology is demonstrated on planar Swift-Hohenberg, capillary-gravity Whitham, and Gray-Scott models, yielding precise stability conclusions, kernel dimensions, and symmetry-induced structure (e.g., radial symmetry) of localized solutions. This framework advances rigorous stability analysis on unbounded domains by combining Fourier-space discretizations, operator-theoretic localization, and interval arithmetic to deliver fully explicit, verifiable results.

Abstract

In this paper, we present a general methodology for investigating the linear stability of localized solutions in PDEs and nonlocal equations on $\mathbb{R}^m$. More specifically, we control the spectrum of the Jacobian $D\mathbb{F}(\tilde{u})$ at a localized solution $\tilde{u}$, enclosing both the eigenvalues and the essential spectrum. Our approach is computer-assisted and is based on a controlled approximation of $D\mathbb{F}(\tilde{u})$ by its Fourier coefficients counterpart on a bounded domain $Ω_d = (-d,d)^m$. We first control the spectrum of the Fourier coefficients operator combining a pseudo-diagonalization and a generalized Gershgorin disk theorem. Then, deriving explicit estimates between the problem on $Ω_d$ and the one on $\mathbb{R}^m$, we construct disks in the complex plane enclosing the eigenvalues of $D\mathbb{F}(\tilde{u})$. Using computer-assisted analysis, the localization of the spectrum is made rigorous and fully explicit. We present applications to the establishment of stability for localized solutions in the planar Swift-Hohenberg PDE, in the planar Gray-Scott model and in the capillary-gravity Whitham equation.

Stability analysis for localized solutions in PDEs and nonlocal equations on $\mathbb{R}^m$

TL;DR

This work develops a computer-assisted framework for linear stability analysis of localized solutions to PDEs and nonlocal equations on , explicitly separating the essential spectrum, given by the symbol range , from discrete eigenvalues. By translating the problem to Fourier coefficients on bounded cubes and employing a pseudo-diagonalization with a generalized Gershgorin theorem, the authors obtain explicit, computable disks that enclose eigenvalues of the Jacobian at the true localized solution , with controlled radii and multiplicities. A two-tier homotopy argument transfers spectral enclosures from a finite-dimensional, periodic surrogate to the full operator , providing rigorous localization away from the essential spectrum. The methodology is demonstrated on planar Swift-Hohenberg, capillary-gravity Whitham, and Gray-Scott models, yielding precise stability conclusions, kernel dimensions, and symmetry-induced structure (e.g., radial symmetry) of localized solutions. This framework advances rigorous stability analysis on unbounded domains by combining Fourier-space discretizations, operator-theoretic localization, and interval arithmetic to deliver fully explicit, verifiable results.

Abstract

In this paper, we present a general methodology for investigating the linear stability of localized solutions in PDEs and nonlocal equations on . More specifically, we control the spectrum of the Jacobian at a localized solution , enclosing both the eigenvalues and the essential spectrum. Our approach is computer-assisted and is based on a controlled approximation of by its Fourier coefficients counterpart on a bounded domain . We first control the spectrum of the Fourier coefficients operator combining a pseudo-diagonalization and a generalized Gershgorin disk theorem. Then, deriving explicit estimates between the problem on and the one on , we construct disks in the complex plane enclosing the eigenvalues of . Using computer-assisted analysis, the localization of the spectrum is made rigorous and fully explicit. We present applications to the establishment of stability for localized solutions in the planar Swift-Hohenberg PDE, in the planar Gray-Scott model and in the capillary-gravity Whitham equation.
Paper Structure (13 sections, 11 theorems, 111 equations, 1 figure)

This paper contains 13 sections, 11 theorems, 111 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A computer-assisted approach
  3. Localization of the eigenvalues
  4. Presentation of the problem
  5. Notations and main assumptions
  6. Fourier coefficients counterparts
  7. Passage between $L^2(\mathbb{R}^m)$ and $\ell^2$, and vice versa
  8. Some spectral properties
  9. Spectrum of $DF(U_0)$ using Gershgorin disks
  10. Control of the spectrum of $D\mathbb{F}(\tilde{u})$ thanks to $DF(U_0)$
  11. Application to the study of stability
  12. The planar Swift-Hohenberg PDE
  13. The unstable square pattern

Key Result

Theorem 1.1

Denote $(\lambda_n)_{n \in \mathbb{Z}^m}$ the eigenvalues of $DF_d(U_0)$. Let $\delta>0$ and let $\mathcal{J}$ be a closed Jordan domain such that $\mathcal{J} \subset \overline{\sigma_\delta}$. Then there exists a positive sequence $(\epsilon_n)_{n \in \mathbb{Z}^m}$, depending only on $\mathcal{J}

Figures (1)

  • Figure :

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.1
  • proof
  • Lemma 3.1
  • ...and 17 more