Extreme waves generated by the interaction of stationary and oscillatory dissipative solitons

We study the interaction of stationary and oscillatory dissipative solitons (DSs) in the framework of two coupled cubic-quintic Ginzburg-Landau equations. Depending on the approach velocity and the cubic cross coupling between counter-propagating DSs, we obtain during the interaction process an amplitude enhancement of up to about a factor of 2.51. For the interaction of oscillatory DSs, we get above a critical value of the cubic cross coupling between counter-propagating DSs a second peak as a function of time during the interaction, an observation apparently not reported before. It emerges that for a range of values of this cubic cross coupling, the second peak can be of a higher amplitude than the first peak and that its structure is frequently more complex than that of the first peak during the interaction. It also turns out that for the case of out-of-phase initial conditions for oscillatory DSs, the second peak is modified and typically reduced in amplitude, while the first peak arising during the interaction is essentially unchanged in size and shape.