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 derivative(t)a = -(i - zeta)a - da(xx) - (1 + i kappa)|a|(2)a + 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 >> 1 and large forcing f >> 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.