# Numerical Investigation of the Dynamics of 'Hot Spots' as Models of Dissipative Rogue Waves
## 5 Conclusions  
The generation and dynamics of rogue waves in a dissipative medium governed by a complex Ginzburg-Landau equation have been investigated. Localized modes in systems with dissipation or gain have been pursued actively over the years. Dissipative defect modes can be found in optical lattices imprinted in focusing or defocusing cubic nonlinear media [39]. Solitons can also be amplified in the presence of external energy input, e.g., an appropriate gain given to a spatial soliton in a quadratic nonlinear medium may lead to redistribution of energy among various modes [40]. From a general perspective in the studies of solitons, insight and understanding in the generation and stability of solitons have been achieved through these nonlinear Schrodinger and complex Ginzburg-Landau equations [29]. A computational approach was adopted here to study the robustness of the Peregrine breather in a dissipative system. Several forms of the gain function were considered, namely, a constant background gain, a Gaussian gain localized in space, and a 'rogue gain' localized in both space and  
Figure 7: Effect of linear gain localized in both space and time: (**a**) generation of rogue waves from continuous wave of the NLSE in the focusing regime \(\sigma=1\); (**b**) small amplitude peak generated in the defocusing regime \(\sigma=-1\). (\(\beta=-0.1\), \(\sigma_{0}=1\), \(\gamma_{2}=0.1\) and \(\alpha\equiv 0\) in both cases). time. Moreover, both linear and cubic gain functions were studied. In most cases, the gain was balanced by a background loss. Although ingenious transformations can provide analytical models of rogue waves, one drawback in performing such transformations is that these variable coefficients may have to fulfill certain conditions [30; 31; 32; 33; 34]. As a result, the exact form of the gain cannot be arbitrary. The merit of a numerical approach is that there is no such restriction on the form of external energy input. The computational studies should enable us to obtain further insight into the problem. For the simplest case where the linear gain or loss is constant throughout the domain, the gain (loss) increases (decreases) the maximum amplitude attained by the rogue wave and shortens (lengthens) the time necessary to reach the maximum. Moreover, a constant background gain will feed energy into other disturbances associated with the rogue wave, leading to multiple peaks with increasing amplitude subsequent to the occurrence of the rogue wave. Secondly, a Gaussian gain localized in space with a constant background loss was considered. Rogue waves gain energy and grow into a pinned mode provided that the maximum linear gain is sufficiently large. However, the rogue wave can be recovered by restricting the localized gain to a finite time interval which we termed a 'rogue gain'. Furthermore, in the focusing regime, such 'rogue gain' can generate trains of 'rogue-wave-like' entities from a continuous wave initial condition. On the other hand, the formation of rogue wave is suppressed if the dissipation is sufficiently strong in the localized region. Both the background and local dissipative effects are essential to destroy the rogue wave. The mere existence of loss in the localized region, where the rogue wave evolves, may not be sufficient to annihilate it and thus the dynamics is indeed intriguing. Both linear and cubic gains were investigated. Analytically, it is then obvious that the latter will overwhelm the former if the amplitude is large enough. When the cubic gain is sufficiently strong, the rogue wave will grow indefinitely and will cause the governing equation and the solution to break down. For scenarios with linear growth only, several peaks with increasing maximum can be formed but the amplitude remains finite. In the present work, the cubic nonlinearity was taken as a positive constant but further exotic combinations of dispersion, nonlinearity, gain, and loss functions will constitute challenging issues for researchers.