Instabilities and splitting of pulses in coupled Ginzburg–Landau equations

We introduce a general system of two coupled cubic complex Ginzburg–Landau (GL) equations that admits exact solitary-pulse (SP) solutions with a stable zero background. Besides representing a class of systems of the GL type, it also describes a dual-core nonlinear optical fiber with gain in one core and losses in the other. By means of systematic simulations, we study generic transformations of SPs in this system, which turn out to be: cascading multiplication of pulses through a subcritical Hopf bifurcation, which eventually leads to a spatio-temporal chaos; splitting of SP into stable traveling pulses; and a symmetry-breaking bifurcation transforming a standing SP into a traveling one. In some parameter region, the Hopf bifurcation is found to be supercritical, which gives rise to stable breathers. Travelling breathers are also possible in the system considered. In a certain parameter region, stable standing SPs, moving permanent-shape ones, and traveling breathers all coexist. In that case, we study collisions between various types of the pulses, which, generally, prove to be strongly inelastic.