Pinning in the Extended Lugiato–Lefever Equation

SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3679-3702, June 2024.
Abstract. We consider a variant of the Lugiato–Lefever equation (LLE), which is a nonlinear Schrödinger equation on a one-dimensional torus with forcing and damping, to which we add a first-order derivative term with a potential [math]. The potential breaks the translation invariance of LLE. Depending on the existence of zeroes of the effective potential [math], which is a suitably weighted and integrated version of [math], we show that stationary solutions from [math] can be continued locally into the range [math]. Moreover, the extremal points of the [math]-continued solutions are located near zeros of [math]. We therefore call this phenomenon pinning of stationary solutions. If we assume additionally that the starting stationary solution at [math] is spectrally stable with the simple zero eigenvalue due to translation invariance being the only eigenvalue on the imaginary axis, we can prove asymptotic stability or instability of its [math]-continuation depending on the sign of [math] at the zero of [math] and the sign of [math]. The variant of the LLE arises in the description of optical frequency combs in a Kerr nonlinear ring-shaped microresonator which is pumped by two different continuous monochromatic light sources of different frequencies and different powers. Our analytical findings are illustrated by numerical simulations.