# Stochastic and Higher-Order Effects on Exploding Pulses
## 3 Exploding Dissipative Solitons and Higher-Order Effects
### Complex Ginzburg-Landau Equation and Short Pulses  
For short pulses but wide enough (\(T_{0}\sim 0.1\) ps), expanding \(|\tilde{A}(\tilde{z},T-T^{\prime})|^{2}\) in a Taylor-series up to first order in \(T^{\prime}\), and considering up to the third-order dispersion \(\beta_{3}\), we can deduce from Equation (6) the following non-integrable quation [38]  
\[\partial_{\tilde{z}}\tilde{A}-\frac{g(\tilde{z},P)}{2}\tilde{A}+\mathrm{i}\, \frac{\beta_{2}}{2}\partial_{T}^{2}\tilde{A}-\frac{\beta_{3}}{6}\partial_{T}^ {3}\tilde{A}=\mathrm{i}\,\gamma\,|\tilde{A}|^{2}\tilde{A}-\frac{1}{\omega_{0} }\partial_{T}(|\tilde{A}|^{2}\tilde{A})-\mathrm{i}\,T_{R}\;\tilde{A}\, \partial_{T}(|\tilde{A}|^{2}), \tag{7}\]  
where \(T_{R}\equiv\int_{0}^{\infty}tR(t)dt\) is the first moment of the Raman response function and \(\int_{0}^{\infty}R(t)dt=1\).  
Figure 3: Phase diagram: Influence of multiplicative noise on exploding solitons for a fixed time scale \(T=10^{4}\). (**a**) Black triangles (\(\triangle\)) denote collapsed states. Red solid circles (\(\bullet\)) stand for exploding dissipative solutions and blue squares (\(\square\)) for oscillating localized solutions. (**b**) Collapse time for exploding dissipative solitons under the influence of multiplicative noise for \(\mu=-0.18\). The average time scale \(T\) for collapse is shown as a function of the noise strength \(\eta\). Black solid circles (\(\bullet\)) represent an average over 50 realizations shown as open squares (\(\square\)).  
One can note that \(|\tilde{A}|^{2}\) has dimensions of power. Therefore, one can define the dimensionless variable \(A\) as \(\tilde{A}=A\sqrt{|\beta_{2}|}/(T_{0}\sqrt{\gamma})\). In the same way, using the dispersion length, the dimensionless variable \(z\): \(\tilde{z}=zT_{0}^{2}/|\beta_{2}|\), and the dimensionless variable \(\tau\): \(T=T_{0}\tau\). Thus, introducing the variables \(A\), \(z\) and \(\tau\) in Equation (7), we can obtain a dimensionless complex CQGLE including three higher-order effects: third-order dispersion, self-steepening, and intrapulse Raman scattering.  
\[\mathrm{i}\,\partial_{z}A+\frac{1}{2}\partial_{\tau}^{2}A+|A|^{2} A-\
u\,|A|^{4}A = \mathrm{i}\,\delta\,A+\mathrm{i}\,\epsilon\,|A|^{2}A+\mathrm{i} \,u\,|A|^{4}A+\mathrm{i}\,\beta\,\partial_{\tau}^{2}A,\] \[+ \mathrm{i}\,\delta_{3}\,\partial_{\tau}^{3}A-\mathrm{i}\,s\, \partial_{\tau}(|A|^{2}A)+\tau_{R}\,A\,\partial_{\tau}(|A|^{2}),\]  
The left side of Equation (8) is nothing but the cubic-quintic non-linear Schrodinger equation, which is conservative. On the right side, the three first terms are related to \(g\), taking account of the linear gain and loss of energy \((\delta)\), and for the non-linear gain or absorption of energy \((\epsilon,\mu)\). The term associated to \(\beta\) plays the role of spectral filtering. The last three terms on the right side are precisely the higher-order effects, which are conservative and whose coefficients \(\delta_{3}\), \(s\), and \(\tau_{R}\) are defined as follows:  
\[\delta_{3}\equiv\frac{\beta_{3}}{6T_{0}|\beta_{2}|};\ s\equiv\frac{1}{\omega_{ 0}T_{0}};\ \tau_{R}\equiv\frac{T_{R}}{T_{0}}. \tag{9}\]