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∂_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.