A bifurcation analysis for the Lugiato-Lefever equation

TL;DR

This paper analyzes the bifurcation structure of the Lugiato-Lefever equation, revealing the existence of various stationary solutions such as localized, periodic, and quasi-periodic waves using bifurcation and normal form theories.

Contribution

It provides a detailed bifurcation analysis of the Lugiato-Lefever equation, identifying conditions for different types of stationary solutions.

Findings

Existence of spatially localized solutions.

Presence of periodic and quasi-periodic solutions.

Identification of codimension 1 bifurcations.

Abstract

The Lugiato-Lefever equation is a cubic nonlinear Schr\"odinger equation, including damping, detuning and driving, which arises as a model in nonlinear optics. We study the existence of stationary waves which are found as solutions of a four-dimensional reversible dynamical system in which the evolutionary variable is the space variable. Relying upon tools from bifurcation theory and normal forms theory, we discuss the codimension 1 bifurcations. We prove the existence of various types of steady solutions, including spatially localized, periodic, or quasi-periodic solutions.