Phase-suppressed hydrodynamics of solitons on constant-background plane wave
Amin Chabchoub, Takuji Waseda, Marco Klein, Stefano Trillo, Miguel Onorato
PPhase-suppressed hydrodynamics of solitons on constant-background plane wave
A. Chabchoub , , , ∗ , T. Waseda , M. Klein , S. Trillo , and M. Onorato , Centre for Wind, Waves and Water, School of Civil Engineering,The University of Sydney, Sydney, New South Wales 2006, Australia ∗ Marine Studies Institute, The University of Sydney, Sydney, New South Wales 2006, Australia Department of Ocean Technology Policy and Environment,Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa, Chiba 277-8563, Japan Institute for Ship Structural Design and Analysis,Hamburg University of Technology, 21073 Hamburg, Germany Department of Engineering, University of Ferrara, 44122 Ferrara, Italy Dipartimento di Fisica, Universit`a degli Studi di Torino, 10125 Torino, Italy and Istituto Nazionale di Fisica Nucleare, INFN, Sezione di Torino, 10125 Torino, Italy
Soliton and breather solutions of the nonlinear Schr¨odinger equation (NLSE) are known to modellocalized structures in nonlinear dispersive media such as on the water surface. One of the conditionsfor an accurate propagation of such exact solutions is the proper generation of the exact initial phase-shift profile in the carrier wave, as defined by the NLSE envelope at a specific time or location.Here, we show experimentally the significance of such initial exact phase excitation during thehydrodynamic propagation of localized envelope solitons and breathers, which modulate a planewave of constant amplitude (finite background). Using the example of stationary black solitonsin intermediate water depth and pulsating Peregrine breathers in deep-water, we show how theselocalized envelopes disintegrate while they evolve over a long propagation distance when the initialphase shift is zero. By setting the envelope phases to zero, the dark solitons will disintegrate into twogray-type solitons and dispersive elements. In the case of the doubly-localized Peregrine breatherthe maximal amplification is considerably retarded; however locally, the shape of the maximalfocused wave measured together with the respective signature phase-shift are almost identical to theexact analytical Peregrine characterization at its maximal compression location. The experiments,conducted in two large-scale shallow-water as well as deep-water wave facilities, are in very goodagreement with NLSE simulations for all cases.
INTRODUCTION
Wave propagation in nonlinear dispersive media are known to be modeled by weakly nonlinear evolution equationssuch as the nonlinear Schr¨odinger equation (NLSE) [1, 2]. Even though the NLSE is restricted in taking into accountonly weak nonlinearities of the wave field and narrow-band processes, several laboratory studies have confirmed itsvalidity to predict the dynamics of stationary and pulsating coherent structures in optics, hydrodynamics and plasma[3–8]. One remarkable feature of the hydrodynamic NLSE is that it takes into account the correct ratio of group andphase velocity even when higher-order effects are at play [9], which are captured in the modified NLSE framework[10, 11].Indeed, the vast and manifold families of exact NLSE solutions allow to quantitatively study the dynamics oflocalized structures, especially, within an experimental framework [12–14], due to given parametrical temporal andspatial distribution of the wave field. When modelling a localized wave train, which exists on a finite background(i.e a carrier wave of constant amplitude and frequency), an exact NLSE solution determines and applies a specificphase-shift profile with respect to the background wave. This fundamental feature is an essential attribute of the lattercorresponding specific NLSE model. Without the correct phase-shift in the initial condition of the NLSE solution,the envelope is expected to diverge from the anticipated trajectory, resulting in the disintegration into several similar or different localized structure.NLSE solutions with finite background differ in the defocusing and focusing regimes, respectively, which describewater wave propagation at different water depths. In the defocusing regime the NLSE supports one-soliton solutionsin the form of stationary dark solitons [15] or multi-soliton dark solutions [16]. The simplest one-soliton solutions arecharacterised by a single parameter that fixes the maximal phase-shift along the envelope profile ranging from 0 to π ,which in turn is related to the darkness of the envelope and to its velocity with respect to the natural group-velocity[17]. The limiting case corresponds to the black solution which has a characteristic shift of π , zero velocity, and totaldarkness at the dip. This shift in the carrier is expected to remain constant throughout the whole propagation in timeand space. Conversely, in the focusing case, the solutions are bright and pulsating and have obviously the propertyof exhibiting a phase profile which varies with the focusing of the envelope.The most fundamental solution, namely the Peregrine breather, which is considered to describe the modulation a r X i v : . [ n li n . PS ] J u l instability for the case of infinite modulation period [18], has a transverse phase profile with maximum shift betweenthe background and the peak elevation that ranges from 0 to π upon evolution, where the maximum value of π occursacross the two transverse zeros of the envelope, and is achieved exactly when the wave-packet reaches its maximalfocusing point [19, 20] (we recall that the Peregrine envelope exhibits also a smooth longitudinal phase profile whichhas no impact on the present experiment, though it may qualitatively affect the dynamics under specific regimes [21]).Here, we will investigate numerically and experimentally the significance of the phase of the initial soliton andbreather hydrodynamics using the black soliton and the Peregrine breather as references. Considering a backgroundfield with input negative (dip) envelope modulation in the defocusing or positive (bump) modulation in the focusingregime, respectively, we show that the suppression of the appropriate phase-shift would engender the disintegrationof the localization. All reported experimental results are in excellent agreement with numerical NLSE simulations.We also discuss the long-term propagation and the universal feature of solitons and breathers deprived from theirdistinctive phase setting. LOCALIZED ENVELOPES AND EXPERIMENTAL SET-UP
Nonlinear waves in intermediate water depth as well as in deep-water can be described by the defocusing and thefocusing time NLSE, respectively. In dimensional form, this evolution equation reads [22] − i (cid:18) ∂ψ∂x + β ∂ψ∂t (cid:19) + 12 β ∂ ψ∂t + γ | ψ | ψ = 0 , (1)where: β = 1 c g , (2) β = − c g ∂ ω∂k , (3) γ = ωk c g sinh ( kh ) (cid:0) cosh (4 kh ) + 8 − ( kh ) (cid:1) − ω (2 kh ) (cid:0) ω cosh ( kh ) + kc g (cid:1) c g (cid:0) gh − c g (cid:1) . (4)Here, g denotes the gravitational acceleration, h is the water depth, k is the wave number of the carrier wave, while thedispersion relation reads ω = √ gk tanh kh and c g = ∂ω∂k is the group velocity of the wave-packets. When kh < . β γ < ψ B ( x, t ) = tanh (cid:20)(cid:114) − β γ (cid:18) t − xc g (cid:19)(cid:21) exp (cid:18) − i β x (cid:19) . (5)This fundamental solution has been observed in a wide range of nonlinear dispersive media, for instance in optics [23],Bose-Einstein condensates [24], plasma [25] and recently also in hydrodynamics [26].In deep-water β γ > ψ P ( x, t ) = − β x β γ (cid:18) t − xc g (cid:19) + β x exp (cid:18) i β x (cid:19) . (6)Note that the maximal wave focusing occurs at x = 0 in this parametrization.The corresponding dimensional spatiotemporal surface elevation, taking into account the second-order Stokes cor-rection, in both, focusing and defocusing cases, is modelled by η ( x, t ) = Re ( ψ ( x, t ) exp [i ( kx − ωt )] + 12 kψ ( x, t ) exp [2i ( kx − ωt )]) . (7)Equation (7) can be used to determine the experimental boundary conditions for the hydrodynamic experiments.These NLSE solutions can be represented as ψ ( x, t ) = A ( x, t ) exp [i ϕ ( x, t )] [27] and hold a phase-shift in the complexenvelope, which is also transmitted to the corresponding water surface elevation signals. In order to physicallyobserve such solutions, it is mandatory to take these phase-shifts into consideration in defining the initial conditionsas prescribed by Eq. (7). In order to ignore this phase-shift in the initial condition, as designated by the exact NLSEsolution while keeping the same initial envelope configuration and geometry, we have to simply replace, at somespecific location x , ψ ( x, t ) by | ψ ( x, t ) | in Eq. (7), which means setting all the phases equal to zero. Consequently, it isexpected from weakly nonlinear theory that the envelope dynamics will be significantly different. For instance in thecase of the black soliton, the condition for the stationary wave propagation is going to be violated. We shall investigatethis type of wave motion experimentally and numerically in detail together with the Peregrine breather for deep-waterconditions. We emphasize that a significant wave propagation in space is required in order to experimentally studythe disparity in the wave propagation.The experiments have been conducted in two large water wave facilities. The first with water of intermediate depthis installed at the Technical University of Berlin, with dimensions of 110 × × and a piston-type wave maker,whereas the deep-water wave flume installed at The University of Tokyo generates the waves by means of flap-typewave maker while its dimensions are 85 × × . The remaining configuration of the facility are similar: anumber a wave gauges are installed along the wave facilities to measure the evolution of the nonlinear wave field anda wave absorber is installed at the end of the facility in order to avoid the reflection of the waves. EXPERIMENTAL AND NUMERICAL RESULTSDark solitons
We will first start with the description of the experiments in the defocusing regime. We recall that in order togenerate dark solitons, we have to satisfy the hydrodynamic dimensionless depth condition kh < .
363 [26].A first set of experiments is carried out with carrier parameters determined by the amplitude a = 0 .
036 m, asteepness of ε = 0 .
08 for a dimensionless depth of kh = 0 .
9. Figure 1 shows the evolution of the black soliton wave(left panel) as well as the case of same envelope depression structure without the initial phase-shift of π [17] (rightpanel), both propagating over a considerable propagation distance of 75 m. Note that at the first gauge, i.e. at thelowest time series in the Fig. 1, the surface elevation are almost indistinguishable by eye in the two cases.Clearly, we can observe a very clean stationary propagation of the black soliton, as expected form NLSE predictions andas already reported by means of a different facility [26]. In the latter work the wave flume and thus the propagationdistance was significantly shorter as in these reported tests. Note, that the present one is to date the longestpropagation of a black soliton reported in water waves and the result proves the robustness of such localized structuresduring their evolution. The right panel in Fig. 1 shows the result of the same type of experiment, however, when the π -phase shift is ignored in the initial conditions the dynamics clearly differs. Indeed, a visible distortion of the wavefield is noticed and the initial envelope dip in the dark wave envelope does not remain localized, but rather leads tofission into two shallower envelope dips that separate from each other with definite (opposite) group velocities.A second example of the same type of experiments is illustrated by choosing different carrier parameters. In thisexample, we slightly modify the degree of nonlinearity by increasing the amplitude of the background to a = 0 .
045 mso that the steepness becomes ε = 0 .
10. The experimental results are depicted in Fig. 2. As in Fig. 1, we observe againan ideal propagation of the black soliton for the initial conditions dictated by the NLSE, whereas without correctinitial phases, the initial dark zero dip fissures. As we increase the steepness, compared to the first experiment, thedegree of nonlinearity is increased as well and the envelope-splitting behaviour is much more pronounced. We point
Time (s) D i s t a n c e f r o m t h e w a v e g e n e r a t o r ( m ) Time (s) D i s t a n c e f r o m t h e w a v e g e n e r a t o r ( m ) FIG. 1. Left: Propagation of black soliton, starting with an exact initial condition, as described from the exact NLSE framework.The amplitude of the carrier is a = 0 .
036 m, the steepness ε = ka = 0 .
08, the still water depth is h = 0 . kh = 0 .
9. Right: Propagation of a similiar initial localized structure having the carrier parameter as inleft panel, however, suppressing the characteristic black soliton phase-shift of π in the initial conditions. Time (s) D i s t a n c e f r o m t h e w a v e g e n e r a t o r ( m ) Time (s) D i s t a n c e f r o m t h e w a v e g e n e r a t o r ( m ) FIG. 2. Left: Propagation of black soliton, starting with an exact initial condition, as described from the exact NLSEframework. Right: Propagation of a similar initial localized structure having the carrier parameter as in (Left), however,ignoring the characteristic black soliton phase-shift of π in the initial conditions. Compared with data in Fig. 1, the waterdepth is the same ( h = 0 . kh = 0 . a = 0 .
045 m, corresponding to higher steepness ε = ak = 0 . out that the observed behavior is consistent with similar observations previously reported in optics [30, 31] and Bose-Einstein condensation [32]. In order to investigate the nature of the fission in these observations, we also conductednumerical NLSE simulations of these latter four cases, as described above and as shown in Fig. 1 and Fig. 2. Thenumerical integration scheme is based on the on the common split-step technique [33, 34]. FIG. 3. Numerical NLSE simulation of the experiments in Figs. 1-2. The top row is relative to the case shown in Fig. 1 (left:soliton input; right: suppressed phase input). The bottom row is relative to the case shown in Fig. 2 (left: soliton input; right:suppressed phase input).
The simulations of wave envelope profiles as depicted in Fig. 3 (top left panel) and (bottom left panel) have beenperformed to demonstrate the accuracy of the numerical scheme predicting the evolution of the stationary blacksoliton and thus, for the sake of accurate interpretation of the numerical results. As can be noticed in such panels,the initial dark soliton envelopes remain indeed stationary. However, by ignoring the π -phase-shift of the carrieraround the envelope-depression-type localization, we evidently observe the break-up of the localized envelope zerodip, seemingly into two gray-type structures with additional dispersive waves (ripples) emitted towards the edges ofthe temporal window. As next step of the study, we compare the measurements recorded from farthest wave gauge,placed 75 m from the wave maker, with the corresponding envelope prediction, extracted from the NLSE simulations.In other words, the final temporal traces in the four cases shown in Fig. 3 superimposed to the farthest water wavetank observations. These comparison results are shown in Fig. 4.Figure 4 shows a very good agreement achieved between these farthest measurements, recorded at a distance of 75m of wave propagation and the corresponding NLSE simulations. This confirms the validity of the NLSE in describingthe dynamics of water waves in finite water depth. We also observe a particular agreement for the cases related tothe fission, when injecting initial conditions ignoring the π -phase-shift. We believe that these prediction results arequite remarkable in view of the very long propagation distance trailed while considering that higher-order effects andexperimental imperfections are always present.Interestingly enough, we point out that the black soliton with suppressed phase shift can break in several pairs(instead of a single pair) of symmetric gray solitons when one enters the semi-classical regime which implies a dominantnonlinearity, i.e. for a given width a much larger wave amplitude compared with the soliton amplitude, or for a given Time (s) W a t e r Su r f a c e ( m ) Time (s) W a t e r Su r f a c e ( m ) Time (s) W a t e r Su r f a c e ( m ) Time (s) W a t e r Su r f a c e ( m ) FIG. 4. Blue continuous lines: wave tank measurements after 75 m of wave propagation. Red dashed lines: NLSE predictionat the same gauge position. Top Left: Last measurement of Fig. 1 (Left) compared with the exact black soliton solution.Top Right: Last measurement of Fig. 1 (Right) compared with numerical NLSE simulations. Bottom Left: Last measurementof Fig. 2 (Left) compared with corresponding numerical NLSE simulations after 75 m of propagation. Bottom Right: Lastmeasurement of Fig. 2 (Right) compared with corresponding numerical NLSE simulations after 75 m of propagation. amplitude a much larger width compared with soliton width [32, 35]. This regime, however, is not accessible in ourexperiment and will be addressed in the future.
Peregrine soliton
As next, we discuss the experiments related to bright doubly-localized (i.e., in x and t ) breather structures. There-fore, we will consider deep-water regime in the following. Due to the expected strong focusing of the wave field, wekeep the steepness parameter small in order to expect a good agreement with weakly nonlinear theory. At the wavemaker we excite either a plane wave with small envelope perturbation as given (in modulus and phase) by Eq. (6) at x = −
25, or the same type of wave, though with suppressed envelope phase. The results for the deep-water case forthe carrier parameter a = 0 .
010 m and ε = 0 .
06 are depicted in Fig. 5.Considering the significant propagation distance of 70 m, we can notice at first glance the growth and decay of thePeregrine breather. On the other hand, when the specific initial phase-shift at the input stage is ignored, a longitudinalretarded wave focusing dynamics is observed. Instead of a maximal focusing expected to occur after 25 m from the
Time (s) D i s t a n c e f r o m t h e w a v e g e n e r a t o r ( m ) Time (s) D i s t a n c e f r o m t h e w a v e g e n e r a t o r ( m ) FIG. 5. Left: Propagation of Peregrine breather, starting with an exact initial condition as described by Eq. (6) at x = −
25 m.The amplitude of the carrier is a = 0 .
010 m, while the steepness is ε = 0 .
06. Right: Propagation of a similar initial breatherstructure for the same carrier parameters as in left panel, however, suppressing the characteristic Peregrine phase-shift imposedin the exact initial condition (Eq. (6), at x = − wave generator, it has been observed around 50 m. Note that, due to the finite number of wave gauges (namely,placed at spatial intervals of 10 m) and therefore the discrete character of water wave field measurements, we arenot able to establish the exact position of maximal wave amplification. The role of initial phase manipulation in thepropagation of an Akhmediev breather had been studied numerically [36]. This type of wave focusing retardationis clearly observed in our corresponding Peregrine breather experiments. We recall that the Peregrine breather isthe limiting case of an Akhmediev breather when the modulation frequency tends to zero. In order to confirm theseobservations, we repeat the same type of experiment with the same carrier amplitude, however, for an increased wavesteepness. This allows a faster evolution of the focusing process due to the increase of the nonlinearity of the wavefield, compared to the previous case. Fig. 6 shows the results of measured wave profiles, assigned to these latter initialconditions of the experiment.The results in Fig. 6 confirm the same wave attributes, already noticed in the observations depicted in Fig. 5. Namely,as the Peregrine breather evolves as expected according to the exact NLSE theory, that is, when exact initial conditionsare injected to the wave maker, the evolution of the wave field when the initial Peregrine phases are not satisfiedshow a similar focusing feature in the evolution, however, noticeably retarded. In this latter case the maximal wavefocusing occurs after 40 m and not 25 m from the wave maker. We also note a distortion of the wave field that mayallow the follow-up focusing of ensuing wave packets. Following these tank observations, numerical NLSE simulationswere performed in order to confirm these physical observations. The latter are shown in Fig. 7.The numerical results in Fig. 7 effectively confirm the experimental observations and the dynamics the correspondingwave field undergoes, as shown in Figs. 5 and 6. In fact, we can clearly annotate the retardation of waves’ maximalamplification. These simulations also allow to quantify the nature of the retardation of maximal wave amplification aswell as the wave envelope distortion that result from the phase-shift prohibition. In the first case the spatial deviationsfor maximal wave focusing are of about 25 m whereas in the second of 15 m. PHASE EVOLUTION ANALYSIS
So far, a comparison of the dark and the Peregrine solitons with the respective cases with equal envelope but phasesset equal to zero has been done only in terms of the envelope amplitude. Here we make an extra effort and we compute
Time (s) D i s t a n c e f r o m t h e w a v e g e n e r a t o r ( m ) Time (s) D i s t a n c e f r o m t h e w a v e g e n e r a t o r ( m ) FIG. 6. Left: Propagation of Peregrine breather, starting with an exact initial condition, as described from exact PeregrineNLSE solution for x = −
25 m. The amplitude of the carrier is a = 0 .
010 m, while the steepness is ε = 0 .
07. Right: Propagationof a similar initial breather structure for the same carrier parameters as in (Left), however with suppression of the characteristicPeregrine phase-shift imposed at x = −
25 m in the initial conditions. the phases of the complex envelope from the time series of the carrier wave. The procedure is based on the theoryof analytic signals and relies on the following procedure: using the inverse discrete Fourier transform, the Fourieramplitudes, ˆ η ( ω ), of the time series η ( t ) of the surface elevation are numerically computed; then, positive frequenciesare multiplied by 2 and negative ones by zero. The discrete Fourier transform is then used to calculate the filteredsignal in physical space. The product of the original time series with the one obtained after filtering is the so-calledanalytic signal, η a ( t ); the complex envelope is then computed by removing the fast oscillation characteristic of thecarrier wave as follows [27]: ψ ( t ) = η a ( t ) exp[ − i ωt ] , (8)with ω the frequency of the carrier wave. The phases ϕ ( t ) are then computed by standard means as ϕ ( t ) = tan − (cid:18) (cid:61) [ ψ ( t )] (cid:60) [ ψ ( t )] (cid:19) . (9)Such procedure has been applied to all our data sets and the results are displayed in Figs. 8 and 9 for the dark solitonand Figs. 10 and 11 for the Peregrine soliton, respectively.In all Figs. 8-11 we superimpose the experimentally retrieved phase temporal profiles, reported as solid red lines,to the corresponding wave elevation (solid blue lines), obviously measured at the same longitudinal location.Specifically, as far as the evolutions shown in Fig. 1 are concerned, we compare in Fig. 8 the phase profiles for thecase of ideal soliton excitation (see top row in Fig. 8) to the case of suppressed phase input (bottom row in Fig. 8).In particular, the exact dark soliton clearly exhibits a phase-shift of π across the vanishing dip of the field. As shown,the soliton phase-shift remains nearly unchanged from the first gauge (see top left panel in Fig. 8) to the last one( x = 75 m, see top right panel in Fig. 8), except for a slight distortion from the flat phase-shift profile of the tailsat x = 75, which is more evident at earlier times ( t <
30 s, top right panel). Conversely, when the input phase-shiftis suppressed, the wave-packet develops an intrinsic phase dynamic, which is already noticeable at the first gauge at x = 5 m (see bottom left panel in Fig. 8), and which evolves at the farthermost gauge at x = 75 m in the profile shownin the bottom right panel in Fig. 8. The latter shows a peak value of the phase-shift (compared with the tails) close tothe expected value of 0 . π from NLSE integration. The phase-shift profile in Fig. 8, bottom right panel, has positive FIG. 7. Numerical NLSE simulation of the experiments in Fig. 5-6. The top row is relative to the case shown in Fig. 5 (left:soliton input; right: suppressed phase input). The bottom row is relative to the case shown in Fig. 6 (left: soliton input; right:suppressed phase input). slope across the left depression envelope amplitude and negative slope across the right depression envelope amplitude,consistently with the phase profile of dark solitons which exhibit negative and positive group-velocity deviation (withrespect to natural group-velocity), respectively. A distortion of the profile (dip in the envelope phase for t <
30 s inbottom right panel) is also present in this case, similar to the black soliton case (top right panel in Fig. 8). Finally,we observe a similar scenario for the phase-shift, also for the larger steepness ε = 0 .
1, as explicitly displayed in Fig. 9,which correspond to the data set reported in Fig. 2. We remark that what we report here constitutes: (i) the firstdirect evidence of the phase dynamics of a hydrodynamic black soliton; (ii) the evidence for the phase dynamics ofthe fissioning wave-packets, which in other areas, such as optics or Bose-Einstein condensation, remains a challengingissue due to the involved fast scales.In Fig. 10 a similar analysis is performed for the experimental data dealing with the Peregrine solutions. The toprow in Fig. 10 shows the Peregrine, close to the input (top left panel; 5 m from the wave generator), and at thefocus point (top right panel; 60 m from the wave generator). A phase shift of π is observed around the zeros of theenvelope amplitude. A slight asymmetry around the peak amplitude, which can be noticed at 60 m is attributed tohigher-order effects, which are likely to cause the formation of the exact zeros on the left and right side of the peak atslightly different propagation lengths (not detectable in the experiment due to discreteness of wave gauge positions).0 FIG. 8. Phase profiles generated by the input black envelope, as in the data set of Fig. 1: blue and red lines display the measuredprofile of envelope amplitude and phase-shift, respectively, taken at the same longitudinal location. Top row, launch of exactblack soliton (as in Fig. 1, left panel). Here left and right panels refer to data from first gauge (5 m from wave generator) andlast gauge (75 m from wave generator, respectively. Bottom row, launch of dark soliton with suppressed phase (as in Fig. 1,right panel). As above, left and right panels refer to data from first gauge at 5 m, and last gauge at 75 m, respectively. Here ε = 0 . Nevertheless, here, the comparison with the phase-suppressed case in the input is even more interesting because, asexplicitly shown in Fig. 10 comparing the top row and the bottom row, a π -phase-shift develops also in the casewhen the phases are set to zero in the initial condition, i.e. even if the phase of the perturbation of the backgroundstrongly deviates from that of the exact solution. This is a remarkable fact that finds its roots in the universality ofthe Peregrine soliton in the dynamics ruled by the NLSE [37, 38]; indeed, in the limit of long perturbations (weakdispersion), it has been shown [37] that the evolution of a wide class of bumps leads to the formation of a local Peregrine soliton. This is a rigorous result in the semiclassical regime, where the reader is referred to [37] for moredetails. Our result seem to indicate the validity of this argument also if we strongly deviates from the semiclassicalregime dominated by the nonlinearity, consistently also with observations in [38].We stress here that the evolution, once reached its maximum amplitude, is locally a Peregrine soliton, i.e. it hasthe same envelope shape as a Peregrine solution, it displays a π -phase-shift at the points where the envelope touchesvanishing amplitude and its amplification factor is equal to three. In the left panel of Fig. 10 the evolution in space ofthe solution with the exact Peregrine initial condition is shown. As can be seen, two π -phase-shifts, one of each sideof the maximum of the envelope, is slowly forming during the evolution. Once the Peregrine solution has reached themaximum amplitude, the π -phase-shifts are clearly visible. Remarkably, a very similar behavior is also exhibited bythe evolution characterized by a larger steepness ( ε = 0 . FIG. 9. Same as in Fig. 8 for the data reported in Fig. 2 corresponding to a larger steepness ε = 0 . DISCUSSION AND CONCLUSION
To summarize, we have discussed experimentally and numerically importance of correct wave phase-shift settingsin the boundary conditions for an experiment for the accurate propagation of localized NLSE envelope solutions,both of stationary- and pulsating-type in finite and infinite water depth, respectively. Two experimental campaignswere performed for different carrier parameters. Two unique large hydrodynamic wave facilities were used in thisexperimental study: the black soliton in finite water depth and the Peregrine model in deep-water. Both sets ofexperiments confirm the validity of weakly nonlinear NLSE theory, namely in the case of exact initial conditions thattake into account the initial phase-shift and in the case when this information in the carrier wave is removed. Inthis latter case the initial localizations exhibit fission behavior into several localized structures of similar kind in verygood agreement with our numerical simulations. To overcome the experimental restrictions that limit the distancefor clean hydrodynamic propagation to 75 and 70 m, respectively, and hence do not allow to establish the nature ofthe asymptotic states, we performed further numerical simulations for the reported cases corresponding to the largestwave steepness values by increasing the distance to 300 m. These are shown in Fig. 12.Indeed, we can observe an interesting type of fission expected from the two types of wave envelope models (blacksoliton and Peregrine breather). Both show similarities and differences in the fission behavior. Even though the basicfeatures of the evolution dynamics can be interpreted as being dominated by dispersive effects only, yet for the darksoliton case we can see that the input localized dip effectively splits into two propagation-invariant gray envelopeswith additional dispersive tails, in agreement with observations over 75 m. On the other hand, the Peregrine brightenvelope shows, at distances substantially exceeding 70 m, a further break-up which produces a cascade of Peregrine2
FIG. 10. Phase profiles of the Peregrine breather waveform shown in Fig. 5: blue and red lines display the measured profile ofenvelope amplitude and phase-shift, respectively, taken at the same longitudinal location. Top row, launch of exact Peregrine(as in Fig. 5, left panel). Here left and right panels refer to data from the first gauge (5 m from the wave generator) andthe closest gauge to maximal amplification point of focusing (30 m from the wave generator). Bottom row, Peregrine withsuppressed phase (as in Fig. 5, right panel). As above, left and right panels refer to data from first gauge ( x = 5 m) and closestgauge to maximum amplification (60 m from wave generator). Here ε = 0 . type localizations that can be seen as limiting case of higher-order modulation instability [39] or universal type ofrogue wave cascade [40], or, in a different language, the decay into a Kuznetsov-Ma soliton [41, 42] coexisting withquasi-Kuznetsov-Ma symmetric pairs at non-zero velocities. We emphasize that further advanced hydrodynamicnumerical simulations associated with inverse scattering analysis [43] as well as motivated experiments in water andother nonlinear media would provide more information about the realistic distribution of such pattern. ∗ [email protected][1] D. J. Benney and A. C. Newell, J. Math. Phys. , 133 (1967).[2] V. E. Zakharov, J. Appl. Mech. Techn. Phys. , 190 (1968).[3] H. C. Yuen and B. M. Lake, Adv. Appl. Mech , 229 (1982).[4] A. Hasegawa, Optical solitons in fibers (Springer, 1989).[5] T. Dauxois and M. Peyrard,
Physics of solitons (Cambridge University Press, 2006).[6] B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, Nature Physics , 790(2010).[7] A. Chabchoub, N. Hoffmann, and N. Akhmediev, Phys. Rev. Lett. , 204502 (2011).[8] H. Bailung, S. Sharma, and Y. Nakamura, Phys. Rev. Lett. , 255005 (2011). FIG. 11. Same as in Fig. 10 for the data reported in Fig. 6 corresponding to a larger steepness ε = 0 . , 341 (1999).[10] K. B. Dysthe, in Proc. R. Soc. Lond. A , Vol. 369 (The Royal Society, 1979) pp. 105–114.[11] A. Goullet and W. Choi, Physics of Fluids , 016601 (2011).[12] N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear pulses and beams (Chapman & Hall, 1997).[13] M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. T. Arecchi, Phys. Rep. , 47 (2013).[14] J. M. Dudley, G. Genty, A. Mussot, A. Chabchoub, and F. Dias, Nature Reviews Physics , 675 (2019).[15] V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP , 823 (1973).[16] A. Fratalocchi, C. Conti, G. Ruocco, and S. Trillo, Phys. Rev. Lett. , 044101 (2008).[17] A. Chabchoub, O. Kimmoun, H. Branger, C. Kharif, N. Hoffmann, M. Onorato, and N. Akhmediev, Phys. Rev. E ,011002 (2014).[18] D. H. Peregrine, J. Australian Math. Soc. Series B. Applied Mathematics , 16 (1983).[19] D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, J. Opt. , 064011 (2013).[20] G. Xu, K. Hammani, A. Chabchoub, J. M. Dudley, B. Kibler, and C. Finot, Phys. Rev. E , 012207 (2019).[21] F. Baronio, S. Chen, and S. Trillo, Opt. Lett. , 427 (2020).[22] H. Hasimoto and H. Ono, J. Phys. Soc. Japan , 805 (1972).[23] A. Weiner, J. Heritage, R. Hawkins, R. Thurston, E. Kirschner, D. Leaird, and W. Tomlinson, Phys. Rev. Lett. , 2445(1988).[24] D. Frantzeskakis, J. Physics A: Mathematical and Theoretical , 213001 (2010).[25] P. K. Shukla and B. Eliasson, Phys. Rev. Lett. , 245001 (2006).[26] A. Chabchoub, O. Kimmoun, H. Branger, N. Hoffmann, D. Proment, M. Onorato, and N. Akhmediev, Phys. Rev. Lett. , 124101 (2013).[27] A. Osborne, Nonlinear Ocean Waves & the Inverse Scattering Transform , Vol. 97 (Academic Press, 2010).[28] N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, Sov. Phys. JETP , 894 (1985).[29] N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, Theor. Math. Phys. (USSR) , 809 (1987).[30] D. Kr¨okel, N. J. Halas, G. Giuliani, and D. Grischkowsky, Phys. Rev. Lett. , 29 (1988). FIG. 12. Left: Propagation of the dark soliton envelope for carrier parameters as in Fig. 3 (bottom right panel) over a muchlarger propagation distance of 300 m. Right: Propagation of the Peregrine breather envelope for carrier parameters as in Fig.7 (bottom right panel) over a much longer propagation distance of 300 m.[31] B. Luther-Davies and Y. Xiaoping, Opt. Lett. , 496 (1992).[32] Z. Dutton, M. Budde, C. Slowe, and L. Vestergaard Hau, Science , 663 (2001).[33] R. Hardin and F. Tappert, SIAM Rev. , 423 (1973).[34] R. A. Fisher and W. K. Bischel, Journal of Applied Physics , 4921 (1975).[35] A. Moro and S. Trillo, Phys. Rev. E , 023202 (2014).[36] M. Erkintalo, G. Genty, B. Wetzel, and J. M. Dudley, Phys. Lett. A , 2029 (2011).[37] M. Bertola and A. Tovbis, Commun. Pure Applied Math. , 678 (2013).[38] A. Tikan, C. Billet, G. El, A. Tovbis, M. Bertola, T. Sylvestre, F. Gustave, S. Randoux, G. Genty, and P. Suret, Phys.Rev. Lett. , 033901 (2017).[39] M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, Phys. Rev. Lett. , 253901(2011).[40] R. Grimshaw and A. Tovbis, Proc. R. Soc. Lond. A , 20130094 (2013).[41] E. A. Kuznetsov, Sov. Phys. Dokl. , 507 (1977).[42] Y.-C. Ma, Stud. Applied Math. , 43 (1979).[43] A. Gelash and R. Mullyadzhanov, Physical Review E101