# Numerical Investigation of the Dynamics of 'Hot Spots' as Models of Dissipative Rogue Waves
## 1 Introduction  
The propagation and dynamics of wave envelopes under the competing physical effects of dispersion, cubic nonlinearity, gain, and loss are applicable to many disciplines in physical science. The complex Ginzburg-Landau equation serves as a useful model and has been studied extensively [1]. Many modes of a variety of properties have been established, ranging from localized pulses to fronts or kink type structures. Recently, rogue waves have been studied intensively as extreme and rare events in physics. Rogue waves will enhance our understanding of physical phenomena in many applied settings, ranging from maritime risk of large amplitude oceanic waves to modern technological advances in optics [2; 3; 4; 5; 6]. The most widely employed theoretical model is probably the nonlinear Schrodinger equation (NLSE), where the Peregrine breather represents a mode localized in both space and time. The simplest canonical form of the NLSE consists of second order dispersion and cubic nonlinearity, and no energy gain or loss is accounted for. Extensions to incorporate the presence of a potential or gain/loss mechanisms have been pursued in the literature. For a single component NLSE with variable dispersion and nonlinearity but with linear gain or loss, a similarity variable transformation can be applied to reduce the problem to one with homogeneous properties. Deducing rogue wave modes then becomes feasible [7]. For the case of a potential trap, one relevant physical situation is Bose-Einstein condensate and the corresponding analytical description is then the Gross-Pitaevskii equation [8]. Both 'bright' (elevation) and 'dark'(depression) rogue wave modes are possible and their interactions with the trapping potential exhibit intriguing relations. Such coupled evolution equations with varying nonlinearities can also be utilized to produce rogue wave modes with trajectories which can be controlled [9]. Besides the NLSE, several other nonlinear evolution systems, e.g., the Lugiato-Lefever equation (LLE) [10], have also been employed as analytical models in studying dissipative rogue waves. LLE is applicable to optical systems with time-delay feedback, e.g., lasers with saturable absorbers. One remarkable difference between the conservative NLSE and the dissipative LLE is the formation of two-dimensional rogue waves. Due to the balance between dissipation and nonlinearity in LLE, formation of two-dimensional rogue waves can overcome collapse dynamics [10; 11; 12]. Similar two-dimensional extreme events can also be generated in dissipative chemical reaction-diffusion systems, which is far from equilibrium [13]. From the experimental viewpoint, unexpectedly large amplitude signals had been actively investigated for optical waveguides. As illustrative examples, large amplitude waves can be detected during soliton explosions for fiber lasers at a critical pump power level [14]. Similar waves can also be induced by long-range chaotic multi-pulse interactions in a fiber laser based on a topological insulator-deposited microfiber photonic device [15]. Indeed, rogue waves can be generated via nonlinear soliton collision in multiple-soliton state of a mode-locked fiber laser through real time spatio-temporal intensity measurements [16]. The complex Ginzburg-Landau equation has been utilized to provide an instructive model for understanding the dynamics of such localized pulses [17]. More precisely,'spiny solitons' can chaotically generate spikes of large amplitude and short duration. These spikes resemble rogue waves and have profiles, spectra, and autocorrelation functions different from other pulses studied earlier in the literature [18]. Such dissipative solitons with extreme spikes may occupy a significant portion of the parameter space of the cubic-quintic complex Ginzburg-Landau equation [17]. Bifurcation diagrams demonstrate the existence of both periodic and chaotic modes in the dynamics of motion. In fluid mechanics, rogue waves have been studied using both experimental and computational approaches as well [19; 20].