Dissipative pulses stabilized by nonlinear gradient terms: A review of their dynamics and their interaction

We study the dynamics as well as the interaction of stable dissipative solitons (DSs) of the cubic complex Ginzburg-Landau equation which are stabilized only by nonlinear gradient (NLG) terms. First we review stationary, periodic, quasi-periodic, and chaotic solutions. Then we investigate sudden transitions to chaotic from periodic and vice versa as a function of one parameter, as well as different outcomes, for fixed parameters, when varying the initial condition. In addition, we present a quasi-analytic approach to evaluate the separation of nearby trajectories for the case of stationary DSs as well as for periodic DSs, both stabilized by nonlinear gradient terms. In a separate section collisions between different types of DSs are reviewed. First we present a concise review of collisions of DSs without NLG terms and then the results of collisions between stationary DSs stabilized by NLG terms are summarized focusing on the influence of the nonlinear gradient term associated with the Raman effect. We point out that both, meandering oscillatory bound states as well as bound states with large amplitude oscillations appear to be specific for coupled cubic complex Ginzburg-Landau equations with a stabilizing cubic nonlinear gradient term.