Stability of bound states of pulses in the Ginzburg-Landau equations

Abstract

We consider bound states of quasisoliton pulses in the quintic Ginzburg-Landau equation and in the driven damped nonlinear Schr\"odinger equation. Using the perturbation theory, we derive dynamical systems describing the interaction between weakly overlapping pulses in both models. Bound states (BS's) of the pulses correspond to fixed points (FP's) of the dynamical system. We found that all the FP's in the quintic model are unstable due to the fact that the corresponding dynamical system proves to have one negative effective mass. Nevertheless, one type of FP, spirals, has an extremely weak instability and may be treated in applications as representing practically stable BS's of the pulses. If one considers an extremely long evolution, the spiral gives rise to a stable dynamical state in the form of an infinite-period limit cycle. For the driven damped model, we demonstrate the existence of fully stable BS's, provided that the amplitude of the driving field exceeds a very low threshold.