# Swift-Hohenberg equation with third order dispersion for optical fiber cavity

## TL;DR

This paper derives a generalized Swift-Hohenberg equation with third-order dispersion to describe the dynamics of a photonic crystal fiber cavity near bistability, analyzing dissipative structures and cavity solitons.

## Contribution

It introduces a novel generalized Swift-Hohenberg equation incorporating third-order dispersion for photonic crystal fiber resonators.

## Key findings

- Derived a real order parameter equation with third-order dispersion.
- Characterized the motion of dissipative structures near instability threshold.
- Numerically demonstrated formation of moving cavity solitons.

## Abstract

We investigate the dynamics of a ring cavity made of photonic crystal fiber and driven by a coherent beam working near the resonant frequency of the cavity. By means of a multiple-scale reduction of the Lugiato-Lefever equation with high order dispersion, we show that the dynamics of this optical device, when operating close to the critical point associated with bistability, is captured by a real order parameter equation in the form of a generalized Swift-Hohenberg equation. A Swift-Hohenberg equation has been derived for several areas of nonlinear science such as chemistry, biology, ecology, optics, and laser physics. However, the peculiarity of the obtained generalized Swift-Hohenberg equation for photonic crystal fiber resonators is that it possesses a third-order dispersion. Based on a weakly nonlinear analysis in the vicinity of the modulational instability threshold, we characterize the motion of dissipative structures by estimating their propagation speed. Finally, we numerically investigate the formation of moving temporal localized structures often called cavity solitons.