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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 $\varepsilon V(x)$. 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. ... mehr


Volltext §
DOI: 10.5445/IR/1000155820
Veröffentlicht am 10.02.2023
Cover der Publikation
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
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