How to excite a rogue wave

Rogue waves are one of those fascinating destructive phenomena in nature that have not been fully explained so far 1–3 . There is a variety of approaches to deal with rogue waves 4,5 . Oceanographers commonly agree that linear theories cannot provide explanations for their existence 6,7 . Only nonlinear theories can explain the dramatic concentration of energy into a single “wall of water” well above the average height of the surrounding waves 3,8,9 . Among nonlinear theories, the most fundamental is based on the nonlinear Schrödinger equation NLSE 6 . If the fundamental approach allows us to give a basic explanation, then it can be extended to more general ones which take into account the two-dimensional nature of the problem, effects of the bottom friction 10 , etc. Thus, starting from the simplest model is essential for understanding the phenomenon. Scientists also agree that the initial process that leads to the formation of a rogue wave is modulation instability 8,11,12 . In common understanding, small periodic perturbations lead to high wave periodic structures during evolution. Generally, perturbations do not necessarily have to be periodic. The appearance of single bumps is more likely when the initial conditions are random or have a specific form. One of the possible formation mechanisms for rogue waves is the creation of Akhmediev breathers ABs 10,13–15 that appear due to modulation instability. Then, larger rogue waves can build up when two or more ABs collide 16 . Rogue waves in the ocean are naturally born from random initial conditions. These conditions can later create a multiplicity of ABs appearing in all possible positions along the surface of the ocean. Double and triple collisions lead to the appearance of either high or giant rogue waves with amplitudes two or three times higher than the average wave crests in the surrounding area. Rogue waves can also be observed in optics when propagating optical radiation in photonic crystal fibers 17,18 . Until now, only randomly created rogue waves have been observed experimentally 17 . The processes here are very similar to what happens in the ocean: we observe all random waves and then select only the highest peaks as prototypes of rogue waves. Clearly, we cannot do much if we leave the creation of rogue waves to chance. On the other hand, preparing special initial conditions could be useful. Thus, the next question that we ask ourselves is the following: what are those specific initial conditions that lead directly to the appearance of rogue waves? Understanding this issue would be useful both in attempts to avoid rogue waves in the way of seafarers and in generating highly energetic pulses emerging from optical fibers. Among previous approaches to this problem, we should mention the ideas of Pelinovsky and co-workers see 19 and the book 20 for a review . Namely, starting the evolution from an initial sharp Gaussian or delta function leads to subsequent dispersion to smaller amplitude waves. Inverting the process in time would generate a high amplitude rogue wave from the initial conditions found in the direct process. However, having analytical expressions for the initial conditions and for the final results is more appealing from both mathematical and experimental points of view. Understanding the nature of rogue waves can lead to a better way to suppress or generate them. In optics, we could then create them systematically and obtain high energy pulses each time when we want them. This work suggests a technique for creating rogue waves out of optical fiber devices. As before 16,21 , we deal here with the standard “selffocusing” NLSE. In dimensionless form, it is given by 22