Stability analysis of numerically exact time-periodic breathers in the Lugiato-Lefever equation: Discrete vs continuum

Abstract

We describe a framework for numerical calculation of time-periodically oscillating (breather) solutions to the discretized Lugiato-Lefever equation (LLE), as well as their linear stability as obtained from numerical Floquet analysis. Compared to earlier approaches, our work allows for the following conclusions: (i) The complete families of solutions are obtained also in regimes of instability; (ii) analysis of Floquet spectra and the corresponding eigenvectors show clearly the nature of the various Hopf and period-doubling bifurcations; (iii) properties of breather solutions to the continuous LLE are connected to corresponding oscillating solutions of the discrete LLE, which is of interest in its own right modeling coupled nonlinear cavities. In particular, we show that the oscillating discrete cavity solitons found in earlier work can be viewed as lattice versions of the continuous LLE breathers, as there is a smooth continuation in parameter space connecting them. Moreover, we confirm the existence of stable breathers at large detunings that was recently observed experimentally and describe their appearance from bifurcations.

Justification of the Lugiato-Lefever Model from a Damped Driven ϕ4 Equation

“…with real valued parameters, which is nothing else but a damped driven nonlinear Schrödinger equation. It models spatiotemporal pattern formation in dissipative, diffractive and nonlinear optical cavities submitted to a continuous laser pump [2][3][4][5][6]. The same model was shown rather immediately to also appear in dispersive optical ring cavities [7].…”

Discrete vector light bullets in coupled χ3 nonlinear cavities

“…Such discrete solitons, both in 1D and in 2D waveguides arrays, have been realized experimentally via Kerr nonlinearity [39,40], quadratic nonlinearity [41], and photorefractive effect [42]. The stability of time-periodically oscillating (breather) solutions in the discretized LLE has been studied in [43,44]. Only quite recently, discrete LBs have been predicted theoretically for the discrete LLE [35].…”

Controlling Microresonator Solitons with the Counter-Propagating Pump

“…We performed both forward and backwards adiabatic scans of the control field frequency and demonstrate non-reciprocity of the soliton-blockade effect (see Section 4). In Section 5, we present equations for perturbations around the solitons, which bare the features specific to the integral nature of the XPM terms and demonstrate the breather states close to the soliton-blockade interval (see, e.g., [21][22][23] for the breather studies in the unidirectional resonators).…”