We study the influence of noise on the spatially localized, temporally regular states (stationary, one frequency, two frequencies) in the regime of anomalous dispersion for the cubic-quintic complex Ginzburg-Landau equation as a function of the bifurcation parameter. We find that noise of a fairly small strength $\eta$ is sufficient to reach a chaotic state with exploding dissipative solitons. That means that noise can induce explosions over a fairly large range of values of the bifurcation parameter $\mu$. Three different routes to chaos with exploding dissipative solitons are found as a function of $\mu$. As diagnostic tools we use the separation to characterize chaotic behavior and the energy to detect spatially localized explosive behavior as a function of time.

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.

Partial annihilation of two counterpropagating dissipative solitons, with only one pulse surviving the collision, has been widely observed in different experimental contexts, over a large range of parameters, from hydrodynamics to chemical reactions. However, a generic picture accounting for partial annihilation is missing. Based on our results for coupled complex cubic-quintic Ginzburg-Landau equations as well as for the FitzHugh-Nagumo equation we conjecture that noise induces partial annihilation of colliding dissipative solitons in many systems.

We investigate the route to exploding dissipative solitons in the complex cubic-quintic Ginzburg-Landau equation, as the bifurcation parameter, the distance from linear onset, is increased. We find for a large class of initial conditions the sequence: stationary localized solutions, oscillatory localized solutions with one frequency, oscillatory localized solutions with two frequencies, and exploding localized solutions. The transition between localized solutions with one and with two frequencies, respectively, is analyzed in detail. It is found to correspond to a forward Hopf bifurcation for these localized solutions as the bifurcation parameter is increased. In addition, we make use of power spectra to characterize all time-dependent states. On the basis of all information available, we conclude that the sequence oscillatory localized solutions with one frequency, oscillatory localized solutions with two frequencies, and exploding dissipative solitons can be interpreted as the analog of the Ruelle-Takens-Newhouse route to chaos for spatially localized solutions.

We investigate a two-dimensional extended system showing chaotic and localized structures. We demonstrate the robust and stable existence of two types of exploding dissipative solitons. We show that the center of mass of asymmetric dissipative solitons undergoes a random walk despite the deterministic character of the underlying model. Since dissipative solitons are stable in two-dimensional systems we conjecture that our predictions can be tested in systems as diverse as nonlinear optics, parametric excitation of granular media and clay suspensions, and sheared electroconvection.

We show the existence of periodic exploding dissipative solitons. These nonchaotic explosions appear when higher-order nonlinear and dispersive effects are added to the complex cubic-quintic Ginzburg-Landau equation modeling fiber soliton lasers. This counterintuitive phenomenon is the result of period-halving bifurcations leading to order (periodic explosions), followed by period-doubling bifurcations leading to chaos (chaotic explosions).