Wesley Perkins: Uniform Stability to Subharmonic Pertubations

We study the linear dynamics of spectrally stable T-periodic stationary solutions of the Lugiato-Lefever equation (LLE), a damped nonlinear Schr¨odinger equation with forcing that arises in nonlinear optics. It is known that such T-periodic solutions are nonlinearly stable to NT-periodic, i.e. subharmonic, perturbations for each N ∈ N with exponential decay rates of the form e −δN t . However, both the exponential rates of decay δN and the allowable size of initial perturbations tend to 0 as N → ∞ so that this result is non-uniform in N and is, in fact, empty in the limit N = ∞. The primary goal of this talk is to introduce a methodology, in the context of the LLE, by which a uniform stability result for subharmonic perturbations may be achieved, at least at the linear level.