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

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 $\Omega_d = (-d,d)^m$. 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 $\Omega_d$ 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.