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The Lugiato-Lefever equation with nonlinear damping caused by two photon absorption

Gärtner, Janina; Mandel, Rainer; Reichel, Wolfgang

In this paper we investigate the effect of nonlinear damping on the Lugiato-Lefever equation
i∂_t a = −(i − ζ)a − da_{xx} − (1 + iκ)|a|^2 a + if
on the torus or the real line. For the case of the torus it is shown that for small nonlinear damping κ > 0 stationary spatially periodic solutions exist on branches that bifurcate from constant solutions whereas all nonconstant solutions disappear when the damping parameter κ exceeds a critical value. These results apply both for normal (d < 0) and anomalous (d > 0) dispersion. For the case of the real line we show by the Implicit Function Theorem that for small nonlinear damping κ > 0 and large detuning ζ >> 1 and large forcing f >> 1 strongly localized, bright solitary stationary solutions exists in the case of anomalous dispersion d > 0. These results are achieved by using techniques from bifurcation and continuation theory and by proving a convergence result for solutions of the time-dependent Lugiato-Lefever equation.

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Volltext §
DOI: 10.5445/IR/1000088289
Veröffentlicht am 13.12.2018
Seitenaufrufe: 30
seit 07.12.2018
Downloads: 6
seit 31.12.2018
Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht
Jahr 2018
Sprache Englisch
Identifikator ISSN: 2365-662X
KITopen-ID: 1000088289
Verlag KIT, Karlsruhe
Umfang 28 S.
Serie CRC 1173 ; 2018/44
Projektinformation SFB 1173/1 (DFG, DFG KOORD, SFB 1173/1 2015)
Schlagworte Lugiato-Lefever equation, bifurcation, continuation, solitons, frequency combs, nonlinear damping, two photon absorption.
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