[{"type":"report","title":"The Lugiato-Lefever equation with nonlinear damping caused by two photon absorption","issued":{"date-parts":[["2018"]]},"DOI":"10.5445\/IR\/1000088289","author":[{"family":"G\u00e4rtner","given":"Janina"},{"family":"Mandel","given":"Rainer"},{"family":"Reichel","given":"Wolfgang"}],"publisher":"KIT","publisher-place":"Karlsruhe","ISSN":"2365-662X","collection-title":"CRC 1173","abstract":"In this paper we investigate the effect of nonlinear damping on the Lugiato-Lefever equation\r\ni\u2202_t a = \u2212(i \u2212 \u03b6)a \u2212 da_{xx} \u2212 (1 + i\u03ba)|a|^2 a + if\r\non the torus or the real line. For the case of the torus it is shown that for small nonlinear damping \u03ba > 0 stationary spatially periodic solutions exist on branches that bifurcate from constant solutions whereas all nonconstant solutions disappear when the damping parameter \u03ba 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 \u03ba > 0 and large detuning \u03b6 >> 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.","keyword":"Lugiato-Lefever equation, bifurcation, continuation, solitons, frequency combs, nonlinear damping, two photon absorption.","number-of-pages":28,"kit-publication-id":"1000088289"}]