Justification of the Lugiato-Lefever model from a damped driven $\phi^4$ equation

TL;DR

This paper justifies the derivation of the Lugiato-Lefever equation from a damped driven $^4$ equation, addressing the mathematical rigor of the envelope approximation in nonlinear optics.

Contribution

It provides a novel and rigorous justification of the envelope approximation for the Lugiato-Lefever equation from a damped driven $^4$ model, considering non-square integrable perturbations.

Findings

First rigorous derivation of the envelope approximation in this context

Decomposition of solutions into background and integrable parts

Addresses non-square integrable perturbation challenges

Abstract

The Lugiato-Lefever equation is a damped and driven version of the well-known nonlinear Schr\"odinger equation. It is a mathematical model describing complex phenomena in dissipative and nonlinear optical cavities. Within the last two decades, the equation has gained a wide attention as it becomes the basic model describing optical frequency combs. Recent works derive the Lugiato-Lefever equation from a class of damped driven $\phi^4$ 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.