Codimension 2 and 3 situations in a ring cavity with elliptically polarized electromagnetic waves (Q2442597)

The work addresses a system of two Lugiato-Lefever equations, i.e., nonlinear-Schrödinger equations including linear loss and external pump fields. The two equations are coupled by cross-phase-modulation (XPM) terms. The system describes the co-propagation of two waves with orthogonal polarizations in a nonlinear pumped lossy laser cavity. The values of the XPM coupling coefficient may correspond to materials with positive- and negative-refractive-index properties. The topic considered in the work is the modulations instability of uniform solutions in the system with periodic boundary conditions. The main problem addressed by the analysis are cases of the degenerate onset of the modulational instability, which takes place simultaneously at two or three different wave numbers of perturbations. Such cases are classified as those with codimension two and three, respectively. The types of the so found simultaneously starting instabilities may be Turing-Turing, Turing-Hopf, or Turing-Hopf-Turing. Parameter regions corresponding to these degeneracies are identified, and numerical simulations of the competing co-starting instability modes are reported.