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

Fiki T. Akbar; Bobby E. Gunara; Hadi Susanto

doi:10.3390/math8050727

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arXiv · 1810.10630

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

Fiki T. Akbar·Bobby E. Gunara·Hadi Susanto

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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 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 $ϕ^4$ 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.

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Fiki T. Akbar, Bobby E. Gunara, Hadi Susanto. 2018-10-24. Justification of the Lugiato-Lefever model from a damped driven $ϕ^4$ equation. https://doi.org/10.3390/math8050727

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