Peregrine Soliton and Breathers in Wave Physics: Achievements and Perspectives

The field of rogue wave physics has been extended well beyond the usual hydrodynamics community during the last decade, in particular, due to the analogy drawn through the one-dimensional nonlinear Schrödinger equation (NLSE) and related nonlinear wave theory. More specifically, breather solutions to the NLSE, whose entire space-time evolution is analytically described, are now considered as the simplest nonlinear prototypes of eminent hydrodynamic rogue waves, which are responsible for many maritime catastrophes. A special focus has been placed on doubly-localized Peregrine-type breathers. Breathers exhibit unique pulsating on non-zero background and localization properties directly linked to the ubiquitous modulation instability phenomenon in one-dimensional wave propagation scenarios.It is worth mentioning that since their discovery in the late 70s breather solutions on finite background and notably those manifesting double localization [1], have remained untested experimentally, thus for almost 30 years, until their emergence into light in 2010 [2]. Following the observation in optics two experimental observations in other distinct field of wave physics, namely in fluids dynamics [3] and plasma [4], followed within a few months after. All of them confirming the controlled generation of doubly-localized and strongly nonlinear Peregrine-type waves in real physical systems. These are now among the highly cited papers in nonlinear wave physics during the last decade. The proof of existence and control of this class of nonlinear waves have been of fundamental significance and are now driving numerous experimental studies in various wave systems, as well as mathematical and engineering developments, worldwide.This multidisciplinary Research Topic commemorates the 10 th anniversary of the observation of Peregrine breather (also commonly called Peregrine soliton or rational breather). A collection of fourteen Original Research, four Brief Research Report, three Mini-Review, and three Review articles are presented, featuring some of the latest advances in theoretical, numerical, and experimental studies of breather waves and modulation instability processes.As an introduction to this Research Topic, we first refer the reader to four articles that review some of the theoretical foundations of breather waves. Karjanto discusses the relationship between periodic first-order breather solutions and their limiting behavior toward the Peregrine breather. Alejo et al. recall the unstable properties of these nonlinear Schrödinger-type breathers, according to a standard definition of stability. The work by Akhmediev focuses on the Peregrine breather as an "elementary particle" of more complicated patterns resulting from its higher-order combination. Such solutions are characterized by 'triangular numbers', defined as the total number of Peregrine wave in a regular pattern, while Osborne elaborates on how to determine the general breather solutions by means of Riemann theta functions.Next, more recent and specific theoretical findings can be found in the following seven articles addressing complex wave propagation problems. The study by Pelinovsky provides a methodology to compute the instability rates for the standing periodic waves and the double-periodic solutions of the NLSE. Ye et al. present exact explicit coupled Peregrine solutions on a periodic-wave background caused by the interference in the vector cubic-quintic nonlinear Schrödinger equation. Uthayakumar et al. review the Peregrine-type solutions appearing under the framework of extended versions of NLSE describing the diverse nonlinear systems. In addition, Peng et al. look into the dynamics of pole trajectories in the complex plane for rational solutions of NLSE and its extensions. El-Tantawy et al. investigate breather waves in a linear damped nonlinear Schrödinger equation. Schober and Islas explore the effects of dissipation and higher-order nonlinearities on the stability of breathers solutions, whilst Singh and Saini examine modulation instability and breathers in a dusty plasma.In parallel, the reader can find four articles with detailed numerical investigations of both fundamental and applied problems, for which no analytical development is yet available. The contribution of Agafontsev and Gelash analyzes the spontaneous emergence of rogue waves for two different initial wave systems, namely the unstable plane wave and the bound-state multi-soliton. Coulibaly et al. report on the possible persistence of Peregrine-type waves in systems outside of the equilibrium, as described by the driven and damped nonlinear Schrödinger equation. From a more applied perspective, Shou and Huang propose a scheme to actively control optical Peregrine breathers in a coherent atomic gas via electromagnetically induced transparency, while Wang et al. investigate the probability of extreme waves in deep crossing random seas in response to the variation of initial spectral bandwidth.Furthermore, the reader will be able to discover eight articles providing recent experimental observations and characterizations of breather or rogue wave dynamics using distinct laboratory facilities based on water waves, ion-acoustic waves, and light waves. Chabchoub et al. show the possible observation of the degenerate soliton interaction, also demonstrated to be a Peregrine breather on the zero-background limit. The brief report by Hsu et al. demonstrates that the Peregrine breather traveling at the free surface of a shear current of slowly varying vorticity may transform into gray solitons. Using a multidipole double plasma device, Pathak analyzes the effect of Landau damping on the evolution of ion acoustic Peregrine breather. Naveau et al. describe a novel optical fiber experimental setup based on a heterodyne optical time-domain reflectometer.Here, breather and recurrence dynamics can be evidenced by monitoring the power and relative phase evolutions of spectral components. Tikan et al. review optical fiber experiments involving either single pulse or partially coherent waves that show the spontaneous and local emergence of Peregrine breather. Xu et al. report the observation of a novel type of breather interaction in telecommunication optical fibers, namely ghost-like breather interaction dynamics, in which two identical breathers propagate with opposite group velocities. The work of Finot stresses and illustrates that great care must be taken when trying to identify the nature of coherent structures in an experimental record. An example of highly peaked structure over a continuous background is shown to have any connection with a Peregrine breather, despite possessing similar key features. Jauberteau et al. shows the possible boosting or taming of spatiotemporal beam instabilities and the possible formation of rogue optical filaments in quadractic crystals.Last but not least, it is worth to highlight the original research of Waseda et al. in a real sea environment, making use of a stereo image sequence of the ocean surface. With the help of a non-linear phase resolving wave model, they evidence a coherent wave group compact in both propagating as well as transverse directions, and oriented obliquely to the propagation direction, similarly to the recently discovered directional solitons and breathers.In conclusion, this Research Topic contains 24 articles devoted to the multifaceted development of ongoing studies in the field of nonlinear Schrödinger breathers and related modulation instability. We strongly hope that this collection will serve as a useful state-of-the-art accomplishments of the research community to date. As such, we anticipate that these contributions will lead to a growing interest, research inspiration and significant advances, which will be benefiting a wide range of theoretical and applied scientists.AUTHOR CONTRIBUTIONS All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.