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Stability of solitary wave solutions in the Lugiato–Lefever equation

Bengel, Lukas 1
1 Institut für Analysis (IANA), Karlsruher Institut für Technologie (KIT)

Abstract:

We analyze the spectral and dynamical stability of solitary wave solutions to the Lugiato-Lefever equation (LLE) on $\mathbb{R}$. Our interest lies in solutions that arise through bifurcations from the phase-shifted bright soliton of the nonlinear Schrödinger equation (NLS). These solutions are highly nonlinear, localized, far-from-equilibrium waves, and are the physical relevant solutions to model Kerr frequency combs. We show that bifurcating solitary waves are spectrally stable when the phase angle satisfies $\theta\in(0,\pi)$, while unstable waves are found for angles $\theta\in(\pi,2\pi)$. Furthermore, we establish orbital asymptotical stability of spectrally stable solitary waves against localized perturbations. Our analysis exploits the Lyapunov-Schmidt reduction method, the instability index count developed for linear Hamiltonian systems, and resolvent estimates.


Volltext §
DOI: 10.5445/IR/1000165621
Veröffentlicht am 15.12.2023
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht/Preprint
Publikationsmonat/-jahr 12.2023
Sprache Englisch
Identifikator ISSN: 2365-662X
KITopen-ID: 1000165621
Verlag Karlsruher Institut für Technologie (KIT)
Umfang 22 S.
Serie CRC 1173 Preprint ; 2023/25
Projektinformation SFB 1173/3 (DFG, DFG KOORD, SFB 1173/3)
Externe Relationen Siehe auch
Schlagwörter stability, bifurcation theory, Lugiato-Lefever equation
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