2018/11/29 by Gärtner, Janina, Mandel, Rainer, Reichel, Wolfgang
#34B15 (Primary) #34C23 #35Q55 #35Q60 (Secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1811.12200
In this paper we investigate the effect of nonlinear damping on the Lugiato-Lefever equation ı∂t a = -(ı-ζ) a - daxx -(1+ıκ)|a|2a +ıf 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.