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

In this paper we investigate the effect of nonlinear damping on the Lugiato-Lefever equation $$\i \partial_t a = -(\i-\zeta) a - da_{xx} -(1+\i\kappa)|a|^2a +\i f$$ 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 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 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.