Periodic waves of the Lugiato-Lefever equation at the onset of Turing instability

We study the existence and the stability of periodic steady waves for a nonlinear model, the Lugiato Lefever equation, arising in optics. Starting from a detailed description of the stability properties of constant solutions, we then focus on the periodic steady waves which bifurcate at the onset of Turing instability. Using a centre manifold reduction, we analyse these Turing bifurcations, and prove the existence of periodic steady waves. This approach also allows us to conclude on the nonlinear orbital stability of these waves for co-periodic perturbations, i.e. for periodic perturbations which have the same period as the wave. This stability result is completed by a spectral stability result for general bounded perturbations. In particular, this spectral analysis shows that instabilities are always due to co-periodic perturbations. This article is part of the theme issue `Stability of nonlinear waves and patterns and related topics'.