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Pinning in the extended Lugiato–Lefever equation

Bengel, Lukas 1; Pelinovsky, Dmitry; Reichel, Wolfgang 1
1 Institut für Analysis (IANA), Karlsruher Institut für Technologie (KIT)

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 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 ε0. 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. ... mehr

Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht/Preprint
Publikationsmonat/-jahr 02.2023
Sprache Englisch
Identifikator ISSN: 2365-662X
KITopen-ID: 1000155820
Verlag Karlsruher Institut für Technologie (KIT)
Umfang 26 S.
Serie CRC 1173 Preprint ; 2023/6
Projektinformation SFB 1173/2 (DFG, DFG KOORD, SFB 1173/2 2019)
Externe Relationen Siehe auch
Siehe auch
Schlagwörter nonlinear Schrödinger equation, bifurcation theory, continuation method
Relationen in KITopen

Volltext §
DOI: 10.5445/IR/1000155820
Veröffentlicht am 10.02.2023
Seitenaufrufe: 148
seit 10.02.2023
Downloads: 72
seit 14.02.2023
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