2023/01/01 by Lukas Bengel, Bengel, Lukas, Dmitry E. Pelinovsky +3 · 1 citation
Physics and Astronomy · #34B15 #34B60 #Advanced Fiber Laser Technologies #Analysis of PDEs (math.AP) #FOS: Mathematics #Mechanical and Optical Resonators #Nonlinear Photonic Systems #Primary: 34C23 #Secondary: 35Q55
paper · pdf · doi:10.48550/arxiv.2302.00311
openalex publication_date 2023/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 εV(x). The potential breaks the translation invariance of LLE. Depending on the existence of zeroes of the effective potential Veff, which is a suitably weighted and integrated version of V, we show that stationary solutions from ε=0 can be continued locally into the range ε\not =0. Moreover, the extremal points of the ε-continued solutions are located near zeros of Veff. We therefore call this phenomenon pinning of stationary solutions. If we assume additionally that the starting stationary solution at ε=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 ε-continuation depending on the sign of Veff' at the zero of Veff and the sign of ε. 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.