2025/05/06 by Matthieu Cadiot, Cadiot, Matthieu · 2 citations
Engineering · Mathematics · #35B35 #35K58 #35P05 #35P15 #47F05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2505.03091
openalex publication_date 2025/05/06 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28
In this paper, we present a general methodology for investigating the linear stability of localized solutions in PDEs and nonlocal equations on ℝm. More specifically, we control the spectrum of the Jacobian D\mathbbF(u) at a localized solution u, enclosing both the eigenvalues and the essential spectrum. Our approach is computer-assisted and is based on a controlled approximation of D\mathbbF(u) by its Fourier coefficients counterpart on a bounded domain Ωd = (-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 Ωd and the one on ℝm, we construct disks in the complex plane enclosing the eigenvalues of D\mathbbF(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.