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Spectral stability of pattern-forming fronts in the complex Ginzburg-Landau equation with a quenching mechanism

We consider pattern-forming fronts in the complex Ginzburg-Landau equation with a traveling spatial heterogeneity which destabilizes, or quenches, the trivial ground state while progressing through the domain. We consider the regime where the heterogeneity propagates with speed $c$ just below the linear invasion speed of the pattern-forming front in the associated homogeneous system. In this situation, the front locks to the interface of the heterogeneity leaving a long intermediate state lying near the unstable ground state, possibly allowing for growth of perturbations. This manifests itself in the spectrum of the linearization about the front through the accumulation of eigenvalues onto the absolute spectrum associated with the unstable ground state. As the quench speed $c$ increases towards the linear invasion speed, the absolute spectrum stabilizes with the same rate at which eigenvalues accumulate onto it allowing us to rigorously establish spectrally stability of the front in $L^2(\mathbb{R})$. The presence of unstable absolute spectrum poses a technical challenge as spatial eigenvalues along the intermediate state no longer admit a hyperbolic splitting and standard tools such as exponential dichotomies are unavailable. Instead, we projectivize the linear flow, and use Riemann surface unfolding in combination with a superposition principle to study the evolution of subspaces as solutions to the associated matrix Riccati differential equation on the Grassmannian manifold. Eigenvalues can then be identified as the roots of the meromorphic Riccati-Evans function, and can be located using winding number and parity arguments.