Interaction process of exploding dissipative solitons

We study the interaction of exploding dissipative solitons (DSs) in the framework of two coupled cubic-quintic Ginzburg-Landau equations. We find several classes of behavior as a function of the cross-coupling between counterpropagating DSs, |c_{r}|, by varying the initial conditions for fixed approach velocity. We use three quantities to characterize the resulting behavior as a function of |c_{r}|: (a) the maximum value of the effective amplitude reached during the interaction process, (b) the time interval between the end of the interaction process between exploding DSs and the start-up of new explosions, and (c) the quantity Ψ related to the effective time of interpenetration obtained by integrating the maximum amplitude over time during the interaction. The differences observed by varying the initial conditions can be traced back to the fact that exploding DSs are intrinsically chaotic as a function of time. Therefore, variations in the phase of the initial conditions or, equivalently, in the initial distance before the start of the collisions, can lead to qualitatively different results. Remarkably one of the possible outcomes, which is particularly sensitive to changes in the initial conditions, is partial annihilation of exploding DSs over a range of |c_{r}|. The fact that either the left or the right pulse survives the collision highlights the deterministic character of the dynamics, even though the system is inherently chaotic. However, for all three types of initial conditions studied a unique behavior emerges as the transition to compound states is approached as a function of |c_{r}| for all three quantities used to characterize the results. For example, the quantity Ψ diverges as the critical value of |c_{r}| is approached.