Pinning in the extended Lugiato-Lefever equation

TL;DR

This paper studies how stationary solutions of a modified Lugiato-Lefever equation can be pinned or localized near zeros of an effective potential, with implications for optical frequency combs in microresonators.

Contribution

It introduces the concept of pinning in the extended Lugiato-Lefever equation and analyzes the stability and continuation of solutions when a potential breaks translation invariance.

Findings

Pinning solutions occur near zeros of the effective potential.

The stability of pinned solutions depends on the sign of the derivative of the effective potential at zeros.

Numerical simulations support the analytical results.

Abstract

We consider a variant of the Lugiato-Lefever equation (LLE), which is a nonlinear Schr\"odinger equation on a one-dimensional torus with forcing and damping, to which we add a first-order derivative term with a potential $\epsilon V(x)$. The potential breaks the translation invariance of LLE. Depending on the existence of zeroes of the effective potential $V_\text{eff}$, which is a suitably weighted and integrated version of $V$, we show that stationary solutions from $\epsilon=0$ can be continued locally into the range $\epsilon\not =0$. Moreover, the extremal points of the $\epsilon$-continued solutions are located near zeros of $V_\text{eff}$. We therefore call this phenomenon \emph{pinning} of stationary solutions. If we assume additionally that the starting stationary solution at $\epsilon=0$ 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 $\epsilon$-continuation depending on the sign of $V_\text{eff}'$ at the zero of $V_\text{eff}$ and the sign of $\epsilon$. 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.