Transition from modulated to exploding dissipative solitons: Hysteresis, dynamics, and analytic aspects

We investigate the properties of and the transition to exploding dissipative solitons as they have been found by Akhmediev's group for the cubic-quintic complex Ginzburg-Landau equation. Keeping all parameters fixed except for the distance from linear onset, mu, we covered a large range of values of mu from very negative values to mu = 0, where the zero solution loses its linear stability. We find, with increasing values of mu, stationary pulses, pulses with rapid oscillations, and pulses modulated with an additional small frequency. The transition to exploding solitons arises via a hysteretic transition involving symmetric and asymmetric pulses with two frequencies. As mu is increased in the regime of exploding solitons, the fraction of symmetric exploding solitons is increasing. At the transition from asymmetric two frequency pulses to exploding solitons, only asymmetric exploding solitons are found. We completed our analysis with an analytic study of the collapse time for the exploding solitons and found good agreement with our numerical results.