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The Lugiato–Lefever Equation with Nonlinear Damping Caused by Two Photon Absorption

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Abstract

In this paper we investigate the effect of nonlinear damping on the Lugiato–Lefever equation \(\mathrm {i}\partial _t a = -(\mathrm {i}-\zeta ) a - da_{xx} -(1+\mathrm {i}\kappa )|a|^2a +\mathrm {i}f\) on the torus or the real line. For the case of the torus it is shown that for small nonlinear damping \(\kappa >0\) stationary spatially periodic solutions exist on branches that bifurcate from constant solutions whereas all nonconstant stationary \(2\pi \)-periodic solutions disappear when the damping parameter \(\kappa \) 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 \(\kappa >0\) and large detuning \(\zeta \gg 1\) and large forcing \(f\gg 1\) strongly localized, bright solitary stationary solutions exist 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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Acknowledgements

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project-ID 258734477—SFB 1173.

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  1. Karlsruhe Institute of Technology, Institute for Analysis, Englerstraße 2, 76131, Karlsruhe, Germany

    Janina Gärtner, Rainer Mandel & Wolfgang Reichel

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  1. Janina Gärtner
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  2. Rainer Mandel
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Correspondence to Rainer Mandel.

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Gärtner, J., Mandel, R. & Reichel, W. The Lugiato–Lefever Equation with Nonlinear Damping Caused by Two Photon Absorption. J Dyn Diff Equat 34, 2201–2227 (2022). https://doi.org/10.1007/s10884-021-09943-x

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