# Stochastic and Higher-Order Effects on Exploding Pulses
## 2 Influence of Additive and Multiplicative Noise on Exploding Dissipative Solitons
### Results  
In [28], we studied the effect of weak additive noise on the spatially localized pulses (either stationary or oscillating with one and two frequencies) concluding that small additive noise is enough to induce exploding dissipative solitons, which are mostly chaotic. Here, we report (Figure 1a) three types of pulses emerging as a function of \(\mu\), including the range where deterministically exploding solitons exist, for varying two decades of \(\eta\) (logarithmic scale). For \(\mu\lesssim 0.22\) and small values of \(\eta\), we find (in agreement with [28]) noisy non-explosive localized solutions, either chaotic or non-chaotic. Large enough values of noise always induce explosions. Non-chaotic explosions are observed at the border separating non-explosive states from exploding solitons. For sufficiently large noise strength \(\eta\), we observe a transition to filling-in, that is, a noisy spatially homogeneous finite amplitude solution. The phase diagram shown in Figure 1a is rather insensitive to the maximum run time \(T\), as we can notice from Figure 1b. There, the filling-in time for the transition from an exploding dissipative soliton to filling-in under the influence of additive noise for \(\mu=-0.25\) is shown. The average time scale \(T\) for filling-in is plotted as a function of the noise strength \(\eta\). Black solid circles represent an average over 50 realizations shown as open squares (\(\square\)). Inspecting this figure, we see that to decrease \(T\) by about one decade, one needs to increase \(\eta\) by 0.01. For \(\mu=-0.25\) and \(\eta=0\), we have a stationary solution. When a small noise (\(\eta\sim 0.003\)) is added to Equation (1), it acts mainly as a perturbation on short length and time scales giving the state a noisy appearance (Figure 2a). When \(\eta\) is increased to \(\sim\)0.03, a perturbation starts growing in the wings. Once this peak has grown, it interacts with the main pulse forming a wide chaotic localized one (this instant is shown as a snapshot in Figure 2b). After this, rapidly the system collapses to a state,similar to the original starting peak. This is what we call _explosion_. For large enough noise (\(\eta\gtrsim 0.06\)), the whole system jumps to a noisy spatially homogeneous finite amplitude solution (Figure 2c). For spatially homogeneous multiplicative noise, where we observe a collective enhancement and suppression (in space) of the amplitude, we report in Figure 3a three types of patterns, as a function of the bifurcation parameter \(\mu\), and \(\eta\) the noise strength: exploding dissipative solitons (red solid circles (\(\bullet\))), oscillating localized states (blue squares (\(\square\))). Either they are not explosive or their frequency of explosions is undetectable for \(T=10^{4}\). In the range of \(\mu\) shown in Figure 3, values of \(\eta\gtrsim 0.3\) induce  
Figure 1: Phase diagram: Effect of additive noise on exploding solitons for a fixed time scale \(T=10^{4}\). (**a**) Black triangles (\(\triangle\)) denote filling-in to the spatially homogeneous finite amplitude solution under the influence of additive noise. Red solid circles (\(\bullet\)) stand for exploding dissipative solutions and blue squares (\(\square\)) for noisy non-explosive localized solutions; (**b**) Filling-in time for the transition from an exploding dissipative soliton to filling-in under the influence of additive noise for \(\mu=-0.25\). The average time scale \(T\) for filling-in is shown as a function of the noise strength \(\eta\). Black solid circles (\(\bullet\)) represent an average of over 50 realizations shown as open squares (\(\square\)). Figure 2: Three snapshots for \(\mu=-0.25\) and additive noise for different values of the noise strength \(\eta\). (**a**) Noisy non-explosive localized solution, \(\eta=0.003\); (**b**) Exploding dissipative solution, \(\eta=0.03\); (**c**) Noisy spatially homogeneous finite amplitude solution, \(\eta=0.2\). collapse (black triangles (\(\triangle\))).