We discuss recent selected developments of the interaction of deterministic dissipative solitons (DSs). Localized patterns considered include stationary DSs, oscillatory DSs with one and two frequencies, as well as exploding DSs.
Starting with quasi-1 D solutions of the cubic-quintic complex Ginzburg-Landau (CGL) equation in their temporally asymptotic state as initial condition, we find as function of the approach velocity and the real part of the cubic interaction of the two counter-propagating envelopes: interpenetration, one compound state made of both envelopes or two compound states. For the latter class not described before both envelopes show DSs superposed at two different locations . As a complement we investigate a large class of initial conditions, which are not temporally asymptotic quasi-1 D DSs. We find that nonunique results for the outcome of collisions are possible: stationary and oscillatory compound states as well as more complex assemblies consisting of quasi-1 D and localized states.
We have demonstrated recently that for three types of non-explosive DSs there is a large number of different outcomes as a result of these collisions including stationary as well as oscillatory bound states and compound states with one and two frequencies. The two most remarkable results are: a) the occurrence of bound states and compound states of exploding DSs as outcome of the collisions of stationary and oscillatory pulses; and b) spatio-temporal disorder due to the creation, interaction and annihilation of DSs for colliding oscillatory DSs as initial conditions. These results will be compared with those of collisions of two counter-propagating exploding DSs. Since exploding DSs have been observed experimentally predominantly in nonlinear optics, we conjecture that our predictions for their
interactions can be tested in laser systems.