Pinning in the extended Lugiato–Lefever equation

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 $\varepsilon V(x)$. 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 $\varepsilon=0$ can be continued locally into the range $\varepsilon \ne 0$. Moreover, the extremal points of the $\varepsilon$-continued solutions are located near zeros of $V_\text{eff}$ . We therefore call this phenomenon pinning of stationary solutions. If we assume additionally that the starting stationary solution at $\varepsilon=0$ 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 $\varepsilon$-continuation depending on the sign of $V'_\text{eff}$ at the zero of $V_\text{eff}$ and the sign of $\varepsilon$. 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. ... mehrOur analytical findings are illustrated by numerical simulations.