Solitary pulses and periodic waves in the parametrically driven complex Ginzburg-Landau equation

摘要

A one-dimensional model of a dispersive medium with intrinsic loss,compensated by a parametric drive, is proposed. It is a combination of thewell-known parametrically driven nonlinear Schr\"{o}dinger (NLS) and complexcubic Ginzburg-Landau equations, and has various physical applications (inparticular, to optical systems). For the case when the zero background isstable, we elaborate an analytical approximation for solitary-pulse (SP)states. The analytical results are found to be in good agreement with numericalfindings. Unlike the driven NLS equation, in the present model SPs feature anontrivial phase structure. Combining the analytical and numerical methods, weidentify a stability region for the SP solutions in the model's parameterspace. Generally, the increase of the diffusion and nonlinear-loss parameters,which differ the present model from its driven-NLS counterpart, lead toshrinkage of the stability domain. At one border of the stability region, theSP is destabilized by the Hopf bifurcation, which converts it into a localizedbreather. Subsequent period doublings make internal vibrations of the breatherchaotic. In the case when the zero background is unstable, hence SPs areirrelevant, we construct stationary periodic solutions, for which a veryaccurate analytical approximation is developed too. Stability of the periodicwaves is tested by direct simulations.