Super Rogue Waves: Observation of a Higher-Order Breather in Water Waves A. Chabchoub, 1, * N. Hoffmann, 1 M. Onorato, 2,3 and N. Akhmediev 4 1 Mechanics and Ocean Engineering, Hamburg University of Technology Eißendorfer Straße 42, 21073 Hamburg, Germany 2 Dipartimento di Fisica, Universita ` degli Studi di Torino, Torino 10125, Italy 3 Istituto Nazionale di Fisica Nucleare, INFN, Sezione di Torino, Torino 10125, Italy 4 Optical Sciences Group, Research School of Physics and Engineering, The Australian National University, Canberra ACT 0200, Australia (Received 17 January 2012; revised manuscript received 28 February 2012; published 29 March 2012) Super rogue waves with an amplitude of up to 5 times the background value are observed in a water- wave tank for the first time. Nonlinear focusing of the local wave amplitude occurs according to the higher-order breather solution of the nonlinear wave equation. The present result shows that rogue waves can also develop from very calm and apparently safe sea states. We expect the result to have a significant impact on studies of extreme ocean waves and to initiate related studies in other disciplines concerned with waves in nonlinear dispersive media, such as optics, plasma physics, and superfluidity. DOI: 10.1103/PhysRevX.2.011015 Subject Areas: Fluid Dynamics, Geophysics, Nonlinear Dynamics I. INTRODUCTION The nature of rogue waves (RWs) has been discussed in the literature for more than a decade [1–4]. A few major approaches have been suggested to explain the high-impact power of these ‘‘monsters of the deep’’ [5]. Theories vary depending on the conditions where these waves appear [6,7]. One remarkable feature of rogue waves is that they appear visibly from nowhere and disappear without a trace. Nonlinear dynamics is one of the approaches that has been successful in predicting the basic features of rogue waves [8–11]. One of the prototypes suggested to model rogue waves is the so-called Peregrine soliton [12,13]. The rea- son is that such a solution describes the growing evolution of a small, localized perturbation of a plane wave with the subsequent peak amplification of 3 above the plane wave. The large-amplitude peak appears just once in evolution (it is doubly localized rather than periodic in space and time). Despite decades of debate [5,6,8,14], only very recently was the fundamental Peregrine breather soliton observed experimentally in fiber optics [15]. Soon after, it was observed in a water-wave tank [16] and a few months later it was observed in multicomponent plasma [17]. These observations proved that the nonlinear approach is fruitful in a description of rogue waves. The nonlinear theory also predicts that, in addition to the unique fundamental Peregrine soliton, there is an infinite hierarchy of higher- order breather solutions with a progressively increasing amplitude [18,19] that are also localized both in space and time. The study of these solutions is crucial in explain- ing the even higher amplitude waves that can be observed in deep-water conditions. Our present experimental study shows that higher-order RWs do exist and can be success- fully generated physically in a water-wave experiment. These observations may have far reaching consequences in our efforts to understand the waves that are, by far, still being characterized as ‘‘mysterious.’’ II. MATHEMATICS AND EXPERIMENT The nonlinear Schro ¨ dinger equation (NLS) is one of the basic approaches used to describe the nonlinear wave evolution in various media [20,21]. In particular, this equa- tion describes gravity waves in deep water [22–24]: i @A @t þ c g @A @x    ! 0 8k 2 0 @ 2 A @x 2  ! 0 k 2 0 2 jAj 2 A ¼ 0; (1) where t and x are the time and longitudinal coordinates, while k 0 and ! 0 ¼ !ðk 0 Þ denote the wave number and the frequency of the carrier wave, respectively, which are connected through the dispersion relation of linear deep-water wave theory, ! 0 ¼ ffiffiffiffiffiffiffi gk 0 p , where g is the gravitational acceleration. The group velocity here is c g : ¼ d! dk ¼ ! 0 2k 0 . The surface elevation ðx; tÞ is related to the NLS vari- able Aðx; tÞ to second order in steepness as follows: ðx; tÞ¼ RefAðx; tÞ exp½iðk 0 x  ! 0 tÞg þ Ref 1 2 k 0 A 2 ðx; tÞ exp½2iðk 0 x  ! 0 tÞg: (2) A dimensionless form of the NLS, i c T þ c XX þ 2j c j 2 c ¼ 0; (3) is obtained from (1) using the rescaled variables: T ¼ ! 0 8 t; X ¼ðx  c g tÞk 0 ¼ xk 0  ! 0 2 t; c ¼ ffiffiffi 2 p k 0 A: (4) * amin.chabchoub@tuhh.de Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- bution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. PHYSICAL REVIEW X 2, 011015 (2012) 2160-3308= 12=2(1)=011015(6) 011015-1 Published by the American Physical Society