Multiplicative noise can lead to the collapse of dissipative solitons

We investigate the influence of spatially homogeneous multiplicative noise on the formation of localized patterns in the framework of the cubic-quintic complex Ginzburg-Landau equation. We find that for sufficiently large multiplicative noise the formation of stationary and temporally periodic dissipative solitons is suppressed. This result is characterized by a linear relation between the bifurcation parameter and the noise amplitude required for suppression. For the regime associated with exploding dissipative solitons we find a reduction in the number of explosions for larger noise strength as well as a conversion to other types of dissipative solitons or to filling-in and eventually a collapse to the zero solution.

Figure 1

The qualitative difference between the effects of additive noise $\delta$ correlated in space and time versus spatially homogeneous multiplicative noise $\delta$ correlated in time is shown for $\mu =-0.4$. While additive noise $\delta$ correlated in space and time ($\eta =0.05$) leads to rapid spatial and temporal random oscillations superposed on a DS (a), spatially homogeneous multiplicative noise ($\eta =0.15$) leads to a spatially correlated (collective) increase and decrease in the pattern amplitude (solid lines), which is random as a function of time (b). The dashed lines are the deterministic stationary DSs in both cases.

Figure 2

The critical strength of multiplicative noise $\eta$ to induce the transition from stationary and periodic dissipative solitons to collapse is plotted as a function of the distance from linear onset $\mu$. We see that there is a linear relation over a large range of values for $\mu$.

Figure 3

Phase diagrams showing the observed patterns as a function of noise strength $\eta$ on the ordinate versus the bifurcation parameter $\mu$ in the regime of exploding dissipative solitons for (a) $T_{max}=2000$ and for (b) $T_{max}=6000$. We show the parameter range $\mu$ from $-0.12$ to $-0.07$ and for $0.28<\eta <0.33$. Solid triangles ($▴$) are representing collapsed states (zero solution), black solid circles ($•$) correspond to exploding dissipative solitons that neither fill-in nor collapse, squares ($\square$) are representing a “two-phase” behavior corresponding either to a filling-in or to an explosive state that is neither filling-in nor collapsing, depending on the chosen initial conditions. Finally the region denoted by the red solid circles ($•$) corresponds to filling-in independent of initial conditions.

Figure 4

The frequency of explosions as a function of the multiplicative noise strength is plotted in (a) on a linear scale and in (b) on a logarithmic scale for four values of the bifurcation parameter: $\mu =-0.12$ denoted as black solid circles ($•$), $\mu =-0.14$ denoted as triangles ($▵$), $\mu =-0.17$ denoted as solid squares ($\blacksquare$), and $\mu =-0.175$ denoted as open circles ($○$). (c) shows the behavior in the vicinity of the transition to exploding dissipative solitons, ${\mu }_c\text{:}\varepsilon =9\times {10}^{-4}$ denoted as squares ($\square$), $\varepsilon =4\times {10}^{-4}$ denoted as solid circles ($•$), $\varepsilon =7\times {10}^{-6}$ denoted as solid squares ($\blacksquare$), and $\varepsilon ={10}^{-6}$ denoted as triangles ($▵$), where $\varepsilon$ denotes the distance from ${\mu }_c\text{:}\varepsilon =\mu -{\mu }_c$.

Figure 5

Possible types of behavior in response to multiplicative noise applied to exploding dissipative solitons (a) $\mu =-0.08,\eta =0.31$ and (b) and (c) $\mu =0.08,\eta =0.29$. (a) shows the collapse to zero at $T\sim 1480$, and (b) and (c) show two possible types of behavior: filling-in for long times ($T\sim 3000$) or persistent explosions. We have plotted in all cases [(a)–(c)] a global variable, namely, an averaged (in space) $R(x,t)$. We note that the applied multiplicative noise (d) shows no special features in the temporal vicinity of collapse or filling-in.