Eur. Phys. J. D (2017) 71: 131
https://doi.org/10.1140/epjd/e2017-80057-2

Regular Article

A bifurcation analysis for the Lugiato-Lefever equation^*

Laboratoire de Mathématiques de Besançon, Univ. Bourgogne Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France

^a e-mail: cyril.godey@univ-fcomte.fr

Received: 27 January 2017
Received in final form: 29 March 2017
Published online: 25 May 2017

Abstract

The Lugiato-Lefever equation is a cubic nonlinear Schrödinger 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.