Rogue waves with rational profiles in unstable condensate and its solitonic model
RRogue waves with rational profiles in unstable condensate and its solitonic model
D. S. Agafontsev , ∗ and A. A. Gelash , P.P. Shirshov Institute of Oceanology of RAS, 117997 Moscow, Russia. Skolkovo Institute of Science and Technology, 121205 Moscow, Russia. Institute of Automation and Electrometry of SB RAS, 630090 Novosibirsk, Russia.
In this brief report we study numerically the spontaneous emergence of rogue waves in (i) mod-ulationally unstable plane wave at its long-time statistically stationary state and (ii) bound-statemulti-soliton solutions representing the solitonic model of this state [Gelash et al, PRL 123, 234102(2019)]. Focusing our analysis on the cohort of the largest rogue waves, we find their practicallyidentical dynamical and statistical properties for both systems, that strongly suggests that the mainmechanism of rogue wave formation for the modulational instability case is multi-soliton interaction.Additionally, we demonstrate that most of the largest rogue waves are very well approximated –simultaneously in space and in time – by the amplitude-scaled rational breather solution of thesecond order.
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I. INTRODUCTION
The phenomenon of rogue waves (RWs) – unusuallylarge waves that appear suddenly from moderate wavebackground – was intensively studied in the recent years.A number of mechanisms were suggested to explain theiremergence, see e.g. the reviews [1–3], with the most gen-eral idea stating that RWs could be related to breather-type solutions of the underlying nonlinear evolution equa-tions [4–6]. Currently, ones of the most popular modelsfor RWs are the Peregrine rational breather [7] and thehigher-order rational breather [8] solutions of the one-dimensional nonlinear Schr¨odinger equation (1D-NLSE)of the focusing type, iψ t + ψ xx + | ψ | ψ = 0 . (1)These rational breathers represent a family of localizedin space and time algebraic solutions, which evolve on afinite background and lead to three-fold, five-fold, seven-fold, and so on, increase in amplitude at the time of theirmaximum elevation. Taking specific and carefully de-signed initial conditions, they were reproduced in well-controlled experiments performed in different physicalsystems [9–13].The 1D-NLSE is integrable in terms of the inversescattering transform (IST), as it allows transformationto the so-called scattering data , which is in one-to-onecorrespondence with the wavefield and, similarly to theFourier harmonics in the linear wave theory, changes triv-ially during the motion. Thanks to its properties, thescattering data can be used to characterize the wavefield.For spatially localized case, the scattering data consists ofthe discrete (solitons) and the continuous (nonlinear dis-persive waves) parts of eigenvalue spectrum, calculatedfor specific auxiliary linear system. For strongly nonlin-ear wavefields, such as the ones where emergence of ra- ∗ Electronic address: [email protected] tional breathers can be expected, the solitons provide themain contribution to the energy [14] and should thereforeplay the dominant role in the dynamics. In particular, ashas been recently demonstrated in [15], the modulation-ally unstable plane wave (the condensate) at its long-timestatistically stationary state can be accurately modeled(in the statistical sense) with a certain soliton gas, de-signed to follow the solitonic structure of the condensate.The latter naturally raises a question of whether there isa difference between the RWs emerging in the two sys-tems. Indeed, in a soliton gas all RWs are multi-solitoninteractions by construction. Hence, if there is no sig-nificant difference, then we can draw a hypothesis thatfor the asymptotic stationary state of the MI (and, pos-sibly, for other strongly nonlinear wavefields) the mainmechanism of RW formation is interaction of solitons.With the present paper, we contribute to the answer onthis question by summarizing our observations of RWs forboth systems. Specifically, we compute time evolution for1000 random realizations of the noise-induced MI of thecondensate and also for 1000 random realizations of 128-soliton solutions modeling the asymptotic state of the MI.For each realization, we analyze one largest RW emergingin the course of the evolution, thus focusing our analysison the largest RWs. For both systems, we observe practi-cally identical dynamical and statistical properties of thecollected RWs. In particular, most of the RWs turn outto be very well approximated – simultaneously in spaceand in time – by the amplitude-scaled rational breathersolution (RBS) of the second order. By measuring the de-viation between the RWs and their fits with RBS as anintegral of the difference in the ( x, t )-space, we find that,in general, the larger the maximum amplitude of the RW,the better its convergence to the RBS of the second or-der (RBS2). The collected RWs for the two systems turnout to be identically distributed by their maximum am-plitude and deviation from the RBS2. Additionally, wedemonstrate that the observed quasi-rational profiles ap-pear already for synchronized three-soliton interactionsand discuss the next steps in the ongoing research of the a r X i v : . [ n li n . PS ] S e p RW origin.Note that in the present paper we consider solutions ofthe 1D-NLSE for three different types of boundary con-ditions: the MI of the condensate for which we use theperiodic boundary, the multi-soliton solutions with van-ishing border conditions and the RBS having constantborder conditions at infinity. Globally, these solutionsare fundamentally different, and the different border con-ditions require application of separate IST techniques,see e.g. [5, 14, 16, 17]. For instance, formally our MIcase corresponds to finite-band scattering data. How-ever, the characteristic widths of the structures (RWs,solitons, RBS) are small compared to the sizes of thestudied wavefields, so that the eigenvalue bands are verynarrow and we neglect their difference from solitons. Thesimilar idea was suggested in [18], where, vice versa, thesoliton gas was considered as a limit of finite-band solu-tions. Effectively, we assume that formation of a RW, asa local phenomenon, represents a similar process for allthree cases of border conditions. As we demonstrate inthe paper, this assumption is supported by the presentedresults, that raises an important problem that we leavefor future studies – explanation of how the three modelsmay exhibit locally similar nonlinear patterns.The paper is organized as follows. In the next Sectionwe describe our numerical methods and initial conditions,and also discuss how we approximate a RW with a RBS.In Section 3 we summarize our observations. Section 4is devoted to discussion, and the final Section 5 containsconclusions.
II. NUMERICAL METHODS
We solve Eq. (1) in a large box x ∈ [ − L/ , L/ L (cid:29)
1, with periodic boundary conditions using the pseudo-spectral Runge-Kutta fourth-order method in adaptivegrid with the grid size ∆ x set from the analysis of theFourier spectrum of the solution; see [19] for detail. As anintegrable equation, the 1D-NLSE conserves an infiniteset of integrals of motion, see e.g. [14]. We have checkedthat the first ten integrals are conserved by our numericalscheme up to the relative errors from 10 − (the firstthree invariants) to 10 − (the tenth invariant) orders.Without loss of generality, the initial conditions for thenoise-induced MI of the condensate can be written as ψ | t =0 = 1 + (cid:15) ( x ) , (2)where (cid:15) ( x ) represents a small initial noise. We use statis-tically homogeneous in space noise with Gaussian Fourierspectrum, (cid:15) ( x ) = a (cid:18) √ πθL (cid:19) / (cid:88) k e − k /θ + iφ k + ikx , (3)where a is the average noise amplitude in the x -space, k = 2 πm/L is the wavenumber, m ∈ Z is integer, θ is the characteristic noise width in the k -space and φ k arerandom phases for each k and each realization of the ini-tial conditions; the average intensity of such noise equalsto a , (cid:104)| (cid:15) | (cid:105) = a . For the numerical experiment, wetake the box of length L = 256 π and small initial noise, a = 10 − , with wide spectrum, θ = 5. Note that theseparameters match those used in [19].To generate the solitonic model of the asymptotic sta-tionary state of the noise-induced MI, we create 128-soliton solutions with the combination of the dressingmethod and 100-digits precision arithmetics as describedin [20]. Each soliton has four parameters: amplitude a j , velocity v j , space position x j and phase Θ j ; here j = 1 , ..., M , M = 128, and the one-soliton solution readsas ψ s ( x, t ) = a exp (cid:20) iv ( x − x ) + i (cid:18) a − v (cid:19) t + i Θ (cid:21) cosh a ( x − x ) − avt √ . Following [15], we distribute soliton amplitudes accordingto the Bohr-Sommerfeld quantization rule, a j = 2 (cid:115) − (cid:18) j − / M (cid:19) , (4)and set soliton velocities to zero, v j = 0, using uniformly-distributed soliton phases Θ j in the interval [0 , π ) anduniformly-distributed space position parameters x j ina narrow interval at the center of the computationalbox. Zero velocities mean that these multi-soliton so-lutions are bound-state. For the 1D-NLSE in normal-ization (1), the Bohr-Sommerfeld rule describes am-plitudes for the bound-state solitonic content of therectangular box wavefield of unit amplitude ψ = 1and width L o = √ πM , calculated with the semi-classical Zakharov-Shabat direct scattering problem, seee.g. [14, 21, 22]. The generated 128-soliton solutionstake values of unity order approximately within the in-terval x ∈ [ − L o / , L o /
2] and remain small outside of it.For more detail on the soliton gas, we refer the readerto [15], where it has been demonstrated that its spectral(Fourier) and statistical properties match those of thelong-time statistically stationary state of the MI.For the soliton gas, we gather the RWs by simulat-ing the time evolution of the 128-soliton solutions in theinterval t ∈ [0 ,
50] and then collecting one largest RWfor each of the 1000 realizations of initial conditions.For time evolution, we use the same pseudo-spectralRunge-Kutta numerical scheme as for the MI of the con-densate, since application of the dressing method withevolving scattering data takes too much computationaltime and provides the same result. The pseudo-spectralscheme uses periodic boundary conditions, so that so-lution ψ ( x, t ) needs to be small near the edges of thecomputational box. We achieve this by taking the boxof length L = 384 √ π , so that our 128-soliton solutionsare of 10 − order near its edges and take values of unityorder, | ψ | ∼
1, only within its central 1 / ≡ L o /L ).For the MI of the condensate, we collect the RWs sim-ilarly, but in the time interval t ∈ [174 , L ( MI ) · ∆ T ( MI ) = L ( SG ) · ∆ T ( SG ) onthe lengths L ( MI,SG ) of the regions where RWs may ap-pear and on the time intervals ∆ T ( MI,SG ) during whichwe wait for the largest RW. For the soliton gas, the col-lected RWs appear approximately in the space interval x ∈ [ − , L ( SG ) = 420. We believe that thisproperty is connected with the behavior of the ensemble-and time-averaged intensity I ( x ) = (cid:104)| ψ ( x, t ) | (cid:105) , which re-mains flat I = 1 inside this interval and starts to deviatefrom unity at its edges. For the MI, the RWs may appearanywhere within the computational box L ( MI ) = 256 π ;together with the observation time for the soliton gascase ∆ T ( SG ) = 50, this yields ∆ T ( MI ) = 26 and the timeinterval t ∈ [174 , ψ (1) p ( x, t ) = e it (cid:20) − it )1 + 2 x + 4 t (cid:21) . (5)The RBS of the second order (RBS2) ψ (2) p is too complexand we refer the reader to [8] where it was first found.Both solutions are localized in space and in time, andevolve on a finite background (the condensate). For ap-proximation of a RW with a RBS, we use the scaling,translation and gauge symmetries of the 1D-NLSE: in-deed, if u ( x, t ) is a solution of Eq. (1), then A e i Θ · u ( χ, τ ),where χ = A ( x − x ), τ = A ( t − t ) and A , Θ ∈ R ,is also a solution. Technically, we detect the maximumamplitude A of a RW together with its position x andtime t of occurrence, and also the phase at maximumamplitude Θ = arg ψ ( x , t ), and then use the scalingcoefficient A = − A/ A = A/ v (cid:54) = 0. To account its influence, one can make a trans-formation u ( x, t ) → e ivx/ − iv t/ · u ( x − vt, t ), which alsoprompts a simple way to find the velocity. Indeed, at thetime of the maximum elevation t , a RBS with zero veloc-ity, v = 0, has constant phase arg ψ ( x, t ) = const in theregion between the two zeros closest to the maximum am-plitude. In contrast, a RBS with nonzero velocity, v (cid:54) = 0,has constant phase slope, arg ψ ( x, t ) − ivx/ III. ROGUE WAVES WITH RATIONALPROFILES
We start this Section with the description of one RWevent for the soliton gas case, and then continue withexamination of RW properties for both systems – thenoise-induced MI close to its asymptotic stationary stateand the soliton gas representing the solitonic model ofthis state.An example of one of the 10 largest RWs collected forthe soliton gas case is shown in Fig. 1. The space pro-file | ψ ( x, t ) | and the phase arg ψ ( x, t ) at the time ofthe maximum elevation t ≈ . x | ψ | – in Fig. 1(B), and the space-time repre-sentation of the amplitude | ψ ( x, t ) | near the RW event –in Fig. 1(C). As indicated in the figures, the space profile | ψ ( x, t ) | and the maximum amplitude max x | ψ | are verywell approximated by the amplitude-scaled RBS2, andthe space-time representation strongly resembles that ofthe RBS2 as well. At the time of the maximum elevation,the RBS2 has four zeros; the RW presented in Fig. 1 alsohas four local minimums that are very close to zero andwhere the phase arg ψ ( x, t ) jumps approximately by π ,see Fig. 1(A). Note that the phase is practically constantbetween the two local minimums closest to the maxi-mum amplitude, as for the velocity-free RBS1 and RBS2.The described phase pattern is sometimes considered asa characteristic feature of RW formation, see [23, 24].The deviation between a RW and its approximationwith a RBS can be measured locally as d (1 , p ( x, t ) = | ψ − ψ (1 , p || ψ | . (6)Fig. 1(D) shows this deviation d (2) p for the RBS2 in the( x, t )-plane: in space – between the two local minimumsclosest to the maximum amplitude x ∈ Ω, and in time– in the interval t − t ∈ [ − . , . d (2) p remains wellwithin 5% for most of the area demonstrated in figure, sothat the RBS2 turns out to be a very good approximationfor the presented RW – simultaneously in space and intime.As an integral measure reflecting the deviation betweena RW and a RBS, one can consider the quantity D (1 , p = (cid:34) (cid:82) x ∈ Ω (cid:82) t +∆ Tt − ∆ T | ψ − ψ (1 , p | dxdt (cid:82) x ∈ Ω (cid:82) t +∆ Tt − ∆ T | ψ | dxdt (cid:35) / . (7) −4 −2 0 2 4−3−2−1012345 x A | ψ || ψ (1) p || ψ (2) p | arg ψ −1 −0.5 0 0.5 1012345 t m a x x | ψ | B max x | ψ | max x | ψ (1) p | max x | ψ (2) p | FIG. 1: (Color on-line)
One of the 10 largest RWs (the coordinate of maximum amplitude is shifted to zero for bettervisualization) for the soliton gas case with time of occurrence t ≈ .
2, maximum amplitude A ≈ . D (2) p ≈ . (A) space profile of the RW | ψ ( x, t ) | at the time t of its maximum elevation, (B) time dependencyof the maximum amplitude max x | ψ | , (C) space-time representation of the amplitude | ψ ( x, t ) | near the RW event, and (D) relative deviation (6) between the wavefield and the fit with the RBS2 in the ( x, t )-plane. In the panel (A) , the thick blackand thin dash-dot red lines indicate the space profile | ψ ( x, t ) | and the phase arg ψ ( x, t ). In the panels (A,B) , the dashedblue and green lines show the fits with the RBS1 and the RBS2, respectively. In the panel (D) , the deviations d (2) p ≥ . Here we choose the region of integration over time t ∈ [ t − ∆ T, t + ∆ T ] from the condition that at t ± ∆ T the RBS2 fit halves its maximum amplitude. Indeed, asdemonstrated below, the collected RWs have maximumamplitudes roughly between 3 . | ψ | > . | t − t | ≤ .
31 and the deviationsare D (1) p ≈ . D (2) p ≈ .
02 for theRBS2.The quantity (7) can be used to assess how well a RWcan be approximated by a RBS. Fig. 2(A) shows the min-imum deviation D p = min {D (1) p , D (2) p } versus the maxi-mum amplitude of the RW A = max | ψ | , for all 1000 RWscollected for the soliton gas case; the RWs better approxi-mated with the RBS1 are indicated with blue squares andthose with the RBS2 – with green circles. For 57 RWsthe best fit turned out to be the RBS1 – the Peregrine breather, while the other 943 RWs were better approxi-mated by the RBS2. According to our observations, thevalue of deviation (7) below 0 .
05 typically means thatthe RW is very well approximated with the correspond-ing RBS; for 0 . (cid:46) D (1 , p (cid:46) . D (1 , p (cid:38) . . .
05; hence, the collectedRWs can be approximated with the RBS1 satisfactoryat best. For the RBS2 we have completely different pic-ture: 768 RWs show deviations from the RBS2 below 0 . .
05. As demonstrated in Fig. 2(A),larger RWs are typically better approximated with theRBS2. In particular, of 143 RWs having maximum am-plitude above 4, 68 have deviation from the RBS2 below0 .
05, and the mean deviation for the entire group of 143RWs is (cid:104)D (2) p (cid:105) ≈ . A D p A RBS RBS A D p B RBS2RBS1 −2 −1 A P ( A ) C soliton gasMI −1 D (2) p P ( D ( ) p ) D soliton gasMI FIG. 2: (Color on-line) (A,B)
Deviation D p = min {D (1) p , D (2) p } between RWs and their best fits with either the RBS1 or theRBS2 versus the maximum amplitude A of the RW: (A) for the soliton gas and (B) for the MI of the condensate close toits statistically stationary state. The blue squares indicate that the best fit is achieved with the RBS1 and the green circles –with the RBS2. (C,D) The PDFs of (C) the maximum amplitude A for all the RWs and (D) the deviation D (2) p for the RWsbetter approximated with the RBS2, for the soliton gas (blue) and the MI of the condensate close to its statistically stationarystate (red). RWs collected close to the statistically stationary stateof the noise-induced MI show the same general propertiesas those for the soliton gas case. Fig. 2(B) demonstratesvery similar “clouds” of RWs approximated with eitherthe RBS1, or the RBS2 on the diagram representing theminimum deviation D p versus the maximum amplitude A . Of the 1000 RWs in total, 36 are better approximatedwith the RBS1 and 964 – with the RBS2. Of those bet-ter approximated with the RBS1, only 3 have deviationsbelow 0 . .
05. Of 964 RWs betterapproximated with the RBS2, 792 have deviations below0 . .
05. In total, 150 RWs have am-plitudes above 4; out of them – 64 have deviation fromthe RBS2 below 0 .
05, and the mean deviation among thegroup of 150 RWs equals to (cid:104)D (2) p (cid:105) ≈ . D (2) p for the RWs better approximated by theRBS2 (green circles in Fig. 2(A,B)) are also nearly iden- tical, Fig. 2(D). Hence, we conclude that the largest RWsfor the two systems show practically identical dynamical(resemblance with the RBS2) and statistical properties.Note that we have repeated simulations for the MI casewith smaller and larger computational boxes and timewindows for collecting the RWs. As a result, we haveobtained the PDF of the maximum amplitude shifted tosmaller or larger amplitudes, respectively. The nearlyperfect correspondence of the two PDFs in Fig. 2(C) ad-ditionally justifies the usage of the simulation parametersdiscussed in the previous Section. IV. DISCUSSION
As we have mentioned in [20], some soliton collisionsat the time of their maximum elevation have space pro-files remarkably similar to those of the RBS1 and theRBS2. Moreover, we have presented an example ofa phase-synchronized three-soliton collision, for whichboth the space profile and the temporal evolution ofthe maximum amplitude were very well approximatedby the RBS2. The solitons in [20] had nonzero velocities;here we modify the two- and three-soliton examples forthe case of zero velocities and examine the local devia-tions d (1 , p ( x, t ) (6) together with the integral deviations D (1 , p (7).Fig. 3 shows an example of three-soliton interactionwith solitons having amplitudes a = 1, a = 1 . a = 2, zero velocities v j = 0, zero space position pa-rameters x j = 0 and, at the initial time t = 0, zerophases Θ j = 0. The space profile | ψ ( x, t ) | at the timeof the maximum elevation t = 0 is remarkably similarto that of the RBS2, and the local deviation d (2) p ( x, t )remains well within 5% for most of the area presentedin the figure as well. The integral deviation (7) equalsto D (2) p ≈ . v j = 0, zero spacepositions parameters x j = 0 and phases Θ j = 0. For thetwo-soliton interactions, the minimum deviations fromthe RBS1 and the RBS2 turned out to be D (1) p ≈ . D (2) p ≈ . (cid:104)D (1) p (cid:105) ≈ . (cid:104)D (2) p (cid:105) ≈ . D (1) p ≈ .
18 and D (2) p ≈ . (cid:104)D (1) p (cid:105) ≈ .
23 and (cid:104)D (2) p (cid:105) ≈ . D (2) p ≈ .
03, that is still very good for com-parison with the RBS2. Hence, we conclude that quasi-rational profiles very similar to that of the RBS2 appearalready for three-soliton interactions, provided that thesolitons are properly synchronized (that is, have coincid-ing positions and phases).We think that the presented elementary three-solitonmodel might provide an explanation of RW formationinside multi-soliton solutions. The most direct way forfuture studies might be a demonstration of a RW for syn-chronized many-soliton solution. Here, however, we facea new question, that is, whether formation of a RW isa collective phenomenon that requires synchronization ofall the solitons, or a “local” event that can be achieved bysynchronizing of a few. Note that even the latter case rep-resents a challenging problem. Indeed, the solitons gen-erating a RW acquire space and phase shifts due to pres-ence of the remaining solitons, that should influence theiroptimal synchronization condition. For remote solitons,the shifts can be computed analytically using the well-known asymptotic formulas, see e.g. [14], which howeverdo not work for our case of a dense soliton gas where allsolitons effectively interact with each other. This leavesus two options: (i) the local numerical synchronization of a small group with “trial and error” method and (ii)the calculation of the generalized space-phase shifts ex-pressions for the closely located solitons.Also note that our study is limited with respect tostatistical analysis of RWs, as we have focused on thelargest RWs, while the “common” RWs may have differ-ent dynamical and statistical properties. Nevertheless,we believe that, since the largest RWs for the two sys-tems show identical properties, the “common” RWs havethe same properties too. Identification of all the RWs ac-cording to the standard criterion A ≥ . V. CONCLUSIONS
In this brief report we have presented our observationsof RWs within the 1D-NLSE model for (i) the modula-tionally unstable plane wave at its long-time statisticallystationary state and (ii) the bound-state multi-solitonsolutions representing the solitonic model of this state.Focusing our analysis on the largest RWs, we have foundtheir practically identical dynamical and statistical prop-erties for both systems. In particular, most of the RWsturn out to be very well approximated – simultaneouslyin space and in time – by the amplitude-scaled rationalbreather solution of the second order (RBS2), and thetwo sets of the collected RWs are identically distributedby their maximum amplitude and deviation from theRBS2. Additionally, we have demonstrated the appear-ance of quasi-rational profiles very similar to that of theRBS2 already for synchronized three-soliton interactions.The main messages of the present paper can be summa-rized as follows. First, a quasi-rational profile very simi-lar to a RBS does not necessarily mean emergence of thecorresponding rational breather, as it can be a manifesta-tion of a multi-soliton interaction. Second, the identicaldynamical and statistical properties of RWs collected forthe two examined systems strongly suggest that the mainmechanism of RW formation should be the same, i.e.,that RWs emerging in the asymptotic stationary state ofthe MI (and, possibly, in other strongly nonlinear wave-fields) are formed as interaction of solitons. However,more study is necessary to clarify how exactly interactionof solitons within a large wavefield may lead to formationof a RW, and we plan to continue this research in futurepublications.
Acknowledgements
Simulations were performed at the Novosibirsk Super-computer Center (NSU). The work of D.S.A was sup-ported was supported by the state assignment of IO RAS,Grant No. 0149-2019-0002. The work of A.A.G was sup-ported by RFBR Grant No. 19-31-60028. −4 −2 0 2 4−3−2−1012345 x A | ψ || ψ (1) p || ψ (2) p | arg ψ FIG. 3: (Color on-line)
Synchronized three-soliton interaction of solitons having amplitudes a = 1, a = 1 . a = 2,zero velocities v j = 0, zero space positions parameters x j = 0 and, at the initial time t = 0, zero phases Θ j = 0: (A) spaceprofile | ψ ( x, t ) | and phase arg ψ ( x, t ) at the time of the maximum elevation t = 0, and (B) relative deviation (6) betweenthe wavefield and the fit with the RBS2 in the ( x, t )-plane. All notations are the same as in Fig. 1(A,D). The deviation (7)from the RBS2 fit equals to D (2) p ≈ . ,603 (2003).[2] K. Dysthe, H. E. Krogstad, and P. Muller, Annu. Rev.Fluid Mech. , 287 (2008).[3] M. Onorato, S. Residori, U. Bortolozzo, A. Montina, andF. T. Arecchi, Phys. Rep. , 47 (2013).[4] K. B. Dysthe and K. Trulsen, Phys. Scr. , 48 (1999).[5] A. Osborne, Nonlinear Ocean Waves and the InverseScattering Transform (Academic Press, 2010).[6] V. I. Shrira and V. V. Geogjaev, J. Eng. Math. , 11(2010).[7] D. H. Peregrine, J. Aust. Math. Soc. Series B, Appl.Math. , 16 (1983).[8] N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo,Phys. Rev. E , 026601 (2009).[9] B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias,G. Genty, N. Akhmediev, and J. M. Dudley, Nat. Phys. , 790 (2010).[10] A. Chabchoub, N. P. Hoffmann, and N. Akhmediev,Phys. Rev. Lett. , 204502 (2011).[11] H. Bailung, S. K. Sharma, and Y. Nakamura, Phys. Rev.Lett. , 255005 (2011).[12] A. Chabchoub, N. Hoffmann, M. Onorato, andN. Akhmediev, Phys. Rev. X , 011015 (2012).[13] A. Chabchoub, N. Hoffmann, M. Onorato, A. Slunyaev,A. Sergeeva, E. Pelinovsky, and N. Akhmediev, Phys.Rev. E , 056601 (2012). [14] S. Novikov, S. V. Manakov, L. P. Pitaevskii, andV. E. Zakharov, Theory of solitons: the inverse scatteringmethod (Springer Science & Business Media, New York,1984).[15] A. Gelash, D. Agafontsev, V. Zakharov, G. El, S. Ran-doux, and P. Suret, Phys. Rev. Lett. , 234102 (2019).[16] E. D. Belokolos, A. I. Bobenko, V. Z. Enolskii, A. R.Its, and V. B. Matveev,
Algebro-geometric approach tononlinear integrable equations (Springer-Verlag, 1994).[17] A. I. Bobenko and C. Klein,
Computational approach toRiemann surfaces (Springer Science and Business Media,2011).[18] G. A. El, A. L. Krylov, S. A. Molchanov, and S. Ve-nakides, Physica D: Nonlinear Phenomena , 653(2001).[19] D. S. Agafontsev and V. E. Zakharov, Nonlinearity ,2791 (2015).[20] A. A. Gelash and D. S. Agafontsev, Phys. Rev. E ,042210 (2018).[21] V. E. Zakharov and A. B. Shabat, Soviet Physics JETP , 62 (1972).[22] Z. V. Lewis, Phys. Lett. A , 99 (1985).[23] D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, J.Opt. , 064011 (2013).[24] G. Xu, K. Hammani, A. Chabchoub, J. M. Dudley, B. Ki-bler, and C. Finot, Phys. Rev. E99