Oscillatory translational instabilities of localized spot patterns in the Schnakenberg reaction-diffusion system on general 2-D domains

For a bounded 2-D planar domain $\Omega$, we investigate the impact of domain geometry on oscillatory translational instabilities of $N$-spot equilibrium solutions for a singularly perturbed Schnakenberg reaction-diffusion system with $\mO(\eps^2) \ll \mO(1)$ activator-inhibitor diffusivity ratio. An $N$-spot equilibrium is characterized by an activator concentration that is exponentially small everywhere in $\Omega$ except in $N$ well-separated localized regions of $\mO(\eps)$ extent. We use the method of matched asymptotic analysis to analyze Hopf bifurcation thresholds above which the equilibrium becomes unstable to translational perturbations, which result in $\mO(\eps^2)$-frequency oscillations in the locations of the spots. We find that stability to these perturbations is governed by a nonlinear matrix-eigenvalue problem, the eigenvector of which is a $2N$-vector that characterizes the possible modes (directions) of oscillation. The $2N\times 2N$ matrix contains terms associated with a certain Green's function on $\Omega$, which encodes geometric effects. For the special case of a perturbed disk with radius in polar coordinates $r = 1 + \sigma f(\theta)$ with \red{$0< \varepsilon \ll \sigma \ll 1$}, $\theta \in [0,2\pi)$, and $f(\theta)$ $2\pi$-periodic, we show that only the mode-$2$ coefficients of the Fourier series of $f$ impact the bifurcation threshold at leading order in $\sigma$. We further show that when $f(\theta) = \cos2\theta$, the dominant mode of oscillation is in the direction parallel to the longer axis of the perturbed disk. Numerical investigations on the full Schnakenberg PDE are performed for various domains $\Omega$ and $N$-spot equilibria to confirm asymptotic results and also to demonstrate how domain geometry impacts thresholds and dominant oscillation modes.