Abstract
The Lugiato-Lefever equation is a damped and driven version of the well-known nonlinear Schrödinger equation. It is a mathematical model describing complex phenomena in dissipative and nonlinear optical cavities. Within the last two decades, the equation has gained much attention as it has become the basic model describing microresonator (Kerr) frequency combs. Recent works derive the Lugiato-Lefever equation from a class of damped driven ϕ 4 equations closed to resonance. In this paper, we provide a justification of the envelope approximation. From the analysis point of view, the result is novel and non-trivial as the drive yields a perturbation term that is not square integrable. The main approach proposed in this work is to decompose the solutions into a combination of the background and the integrable component. This paper is the first part of a two-manuscript series.
Highlights
The Lugiato-Lefever equation is given by [1] iAτ = − Aξξ − iα A−| A|2 A + F, 2ω ξ ∈ R, τ ≥ 0, (1)with real valued parameters, which is nothing else but a damped driven nonlinear Schrödinger equation
Lugiato-Lefever equation has raised a wide interest following its recent successful experimental application in the study of broadband microresonator-based optical frequency combs [8,9], that has opened applicative avenues
Ferré et al [14] showed that the dynamics of the Lugiato-Lefever equation can be obtained from a driven dissipative sine-Gordon model
Summary
With real valued parameters, which is nothing else but a damped driven nonlinear Schrödinger equation. Ferré et al [14] showed that the dynamics of the Lugiato-Lefever equation can be obtained from a driven dissipative sine-Gordon model. The former equation is a single envelope approximation, that is, modulation equation, of the latter. The ’hardening’ case λ > 0 will be discussed in the second part of this paper series, whose results can be extended to the sine-Gordon equation This equation belongs to the class of nonlinear Klein-Gordon models that have broad applications [15]. We have considered the reduction of a Klein-Gordon equation with external damping and drive into a damped driven discrete nonlinear Schrödinger equation [27].
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