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 - d a xx - (1 + i κ) | 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 κ > 0 stationary spatially periodic solutions exist on branches that bifurcate from constant solutions whereas all nonconstant stationary 2 π -periodic 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 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. [ABSTRACT FROM AUTHOR]