Numerical Investigation of the Dynamics of ‘Hot Spots’ as Models of Dissipative Rogue Waves

Abstract

In this paper, the effect of gain or loss on the dynamics of rogue waves is investigated by using the complex Ginzburg-Landau equation as a framework. Several external energy input mechanisms are studied, namely, constant background or compact Gaussian gains and a ‘rogue gain’ localized in space and time. For linear background gain, the rogue wave does not decay back to the mean level but evolves into peaks with growing amplitude. However, if such gain is concentrated locally, a pinned mode with constant amplitude could replace the time transient rogue wave and become a sustained feature. By restricting such spatially localized gain to be effective only for a finite time interval, a ‘rogue-wave-like’ mode can be recovered. On the other hand, if the dissipation is enhanced in the localized region, the formation of rogue wave can be suppressed. Finally, the effects of linear and cubic gain are compared. If the strength of the cubic gain is large enough, the rogue wave may grow indefinitely (‘blow up’), whereas the solution under a linear gain is always finite. In conclusion, the generation and dynamics of rogue waves critically depend on the precise forms of the external gain or loss.