Eckhaus instability in the Lugiato-Lefever model

We study theoretically the primary and secondary instabilities undergone by the stationary periodic patterns in the Lugiato-Lefever equation in the focusing regime. Direct numerical simulations in a one-dimensional periodic domain show discrete changes of the periodicity of the patterns emerging from unstable homogeneous steady states. Through continuation methods of the steady states we reveal that the system exhibits a set of wave instability branches. The organisation of these branches suggests the existence of an Eckhaus scenario, which is characterized in detail by means of the derivation of their amplitude equation in the weakly nonlinear regime. The continuation in the highly nonlinear regime shows that the furthest branches become unstable through a Hopf bifurcation.