A bifurcation analysis for the Lugiato-Lefever equation

Abstract

The Lugiato-Lefever equation is a cubic nonlinear Schrödinger equation, including damping, detuning and driving, which arises as a model in nonlinear optics. We study the existence of stationary waves which are found as solutions of a four-dimensional reversible dynamical system in which the evolutionary variable is the space variable. Relying upon tools from bifurcation theory and normal forms theory, we discuss the codimension 1 bifurcations. We prove the existence of various types of steady solutions, including spatially localized, periodic, or quasiperiodic solutions.

Localized structures in dispersive and doubly resonant optical parametric oscillators

“…In terms of spatial dynamics this point corresponds to a reversible Takens-Bogdanov bifurcation [51,55], and a weakly nonlinear solution of the form A − A + ∼ asech 2 (bx) can be obtained as already done in Refs. [64][65][66].…”

A Priori Bounds and Global Bifurcation Results for Frequency Combs Modeled by the Lugiato--Lefever Equation

“…Unlike [8,9] we consider (1.1) on certain spaces of 2π-periodic functions. We obtain a very rich bifurcation picture which is not limited to local considerations as in [8,9,20]. In Theorem 1 and Theorem 2 we find a priori bounds and uniqueness results which allow us to show that (a) nonconstant solutions of (1.1) only occur in the range sign(d)ζ ∈ [ζ * , ζ * ], (b) nonconstant solutions of (1.1) satisfy a ∞ + |ζ| ≤ C. Here, the values ζ * , ζ * and C are explicit and only depend on the parameters f, d. We begin with our results concerning pointwise a priori bounds for solutions of (1.1) in terms of the parameters ζ, d, f .…”

“…The case d < 0 corresponds to normal dispersion whereas d > 0 is called the anomalous regime, cf. [8,9]. The loss of power due to radiation and waveguide coupling is modeled by the damping term −iâ k (t).…”

On the transition to secondary Kerr combs in whispering-gallery mode resonators

“…We expect these results to provide a deeper understanding of the dynamics of many other dissipative structures in WGM resonators [40,41]. In the light of previous research works that already unveiled key mathematical properties of the LLE equation [42][43][44][45], we also expect our analysis to lead the way to a stability enhancement of these patterns, for the benefit of the many targeted applications [4][5][6][7][8][9].…”