Abstract:

Pattern-forming fronts invading a destabilized ground state in spatially homogeneous systems are generally considered unstable as any perturbation ahead of the front grows exponentially in time due to the instability of the ground state. Nevertheless, pattern-forming fronts are observed in various spatially inhomogeneous settings such as light-sensing reaction-diffusion systems, directional solidification of crystals or ion beam milling. In these settings the unstable state is only established in the wake of the progressing heterogeneity after which patterns start to nucleate. Consequently, perturbations cannot grow far ahead of the interface of the pattern-forming front. This begs the question of whether stability can be rigorously established. In this talk, I answer this question in the affirmative by presenting a stability result for pattern-forming fronts against -perturbations in the spatially inhomogeneous complex Ginzburg-Landau equation. A technical challenge is posed by the presence of unstable absolute spectrum which prohibits the use of standard tools such as exponential dichotomies. Instead, we projectivize the linear flow and study the associated matrix Riccati equation on the Grassmannian manifold. Eigenvalues can then be identified as the roots of the meromorphic Riccati-Evans function. This is joint work with Ryan Goh (Boston University).