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We review the study of rogue waves and related instabilities in optical and oceanic environments, with particular focus on recent experimental developments. In optics, we emphasize results arising from the use of real-time measurement…

Fluid Dynamics · Physics 2019-12-24 J. M. Dudley , G. Genty , A. Mussot , A. Chabchoub , F. Dias

Optical rogue waves are rare yet extreme fluctuations in the value of an optical field. The terminology was first used in the context of an analogy between pulse propagation in optical fibre and wave group propagation on deep water, but has…

Optics · Physics 2014-10-14 John M. Dudley , Frédéric Dias , Miro Erkintalo , Goëry Genty

Rogue waves are solitary waves with extreme amplitudes, which appear to be a ubiquitous phenomenon in nonlinear wave propagation, with the requirement for a nonlinearity being their only unifying characteristics. While many mechanisms have…

We consider the \emph{focusing} nonlinear Schr\"odinger equation posed on the one dimensional line, with nonzero background condition at spatial infinity, given by a homogeneous plane wave. For this problem of physical interest, we study…

Analysis of PDEs · Mathematics 2017-06-07 Claudio Muñoz

The one-dimensional focusing nonlinear Schrodinger equation (NLSE) on an unstable condensate background is the fundamental physical model, that can be applied to study the development of modulation instability (MI) and formation of rogue…

Exactly Solvable and Integrable Systems · Physics 2018-02-14 A. A. Gelash

We present exact solutions for rogue waves arising on the background of periodic waves in the focusing nonlinear Schrodinger equation. The exact solutions are obtained by characterizing the Lax spectrum related to the periodic waves and by…

Exactly Solvable and Integrable Systems · Physics 2020-01-06 Jinbing Chen , Dmitry E. Pelinovsky , Robert E. White

Based on the soliton solution on a continuous wave background for an integrable Hirota equation, the reduction mechanism and the characteristics of the Peregrine rogue wave in the propagation of femtosecond pulses of optical fiber are…

Optics · Physics 2012-05-09 Guangye Yang , Lu Li , Suotang Jia

We considered the modulational instability of continuous-wave backgrounds, and the related generation and evolution of deterministic rogue waves in the recently introduced parity-time (PT)-symmetric system of linearly-coupled nonlinear…

Optics · Physics 2013-07-17 Yu. V. Bludov , R. Driben , V. V. Konotop , B. A. Malomed

We report optical fiber experiments allowing to investigate integrable turbulence in the focusing regime of the one dimensional nonlinear Schr\"odinger equation (1D-NLSE). Our experiments are very similar in their principle to water tank…

Pattern Formation and Solitons · Physics 2015-04-16 Pierre Walczak , Stéphane Randoux , Pierre Suret

We demonstrate a possibility to make rogue waves (RWs) in the form of the Peregrine soliton (PS) and Kuznetsov-Ma breathers (KMBs) effectively stable objects, with the help of properly defined dispersion or nonlinearity management applied…

Pattern Formation and Solitons · Physics 2018-03-14 J. Cuevas-Maraver , B. A. Malomed , P. G. Kevrekidis , D. J. Frantzeskakis

We present exact rational solution for a modified nonlinear Schr$\ddot{o}$dinger equation that takes into account quintic nonlinearity and nonlinear dispersion corrections to the cubic nonlinearity, which could be used to describe rogue…

Pattern Formation and Solitons · Physics 2014-07-15 Li-Chen Zhao , Chong Liu , Zhan-ying Yang

We present a brief discussion on the nonlinear Schr{\"o}dinger equation for modeling the propagation of the deep-water wavetrains and a discussion on its doubly-localized breather solutions that can be connected to the sudden formation of…

Exactly Solvable and Integrable Systems · Physics 2013-01-08 Nikolay K. Vitanov , Amin Chabchoub , Norbert Hoffmann

Rogue wave patterns in the nonlinear Schr\"{o}dinger equation are analytically studied. It is shown that when an internal parameter in the rogue waves (which controls the shape of initial weak perturbations to the uniform background) is…

Exactly Solvable and Integrable Systems · Physics 2021-01-06 Bo Yang , Jianke Yang

Specific solutions of the nonlinear Schr\"odinger equation, such as the Peregrine breather, are considered to be prototypes of extreme or freak waves in the oceans. An important question is, whether these solutions also exist in the…

Pattern Formation and Solitons · Physics 2019-09-27 Leo Dostal , Marten Hollm , Edwin Kreuzer

In this work, based on the recently proposed (Phys. Rev. Lett. 110 (2013) 064105) continuous nonlocal nonlinear Schr\"odinger system with parity-time symmetric Kerr nonlinearity (PTNLSE), a numerical investigation has been carried out for…

Optics · Physics 2017-11-22 Samit Kumar Gupta

Optical rogue waves and its variants have been studied quite extensively in the context of optical fiber in recent years. It has been realized that dispersion management in optical fiber is experimentally much more feasible compared to its…

Modulational instability has been used to explain the formation of breather and rogue waves qualitatively. In this paper, we show modulational instability can be used to explain the structure of them in a quantitative way. We develop a…

Exactly Solvable and Integrable Systems · Physics 2017-04-04 Liming Ling , Li-Chen Zhao

General high-order rogue waves in the nonlinear Schroedinger equation are derived by the bilinear method. These rogue waves are given in terms of determinants whose matrix elements have simple algebraic expressions. It is shown that the…

Exactly Solvable and Integrable Systems · Physics 2015-05-30 Yasuhiro Ohta , Jianke Yang

We study the existence and properties of rogue wave solutions in different nonlinear wave evolution models that are commonly used in optics and hydrodynamics. In particular, we consider Fokas-Lenells equation, the defocusing vector nolinear…

Exactly Solvable and Integrable Systems · Physics 2015-06-23 Fabio Baronio , Shihua Chen , Philippe Grelu , Stefan Wabnitz , Matteo Conforti

We study numerically the properties of (statistically) homogeneous soliton gas depending on soliton density (proportional to number of solitons per unit length) and soliton velocities, in the framework of the focusing one-dimensional…

Pattern Formation and Solitons · Physics 2018-10-24 A. A. Gelash , D. S. Agafontsev