Observation of modulation instability and rogue breathers on stationary periodic waves

We present both theoretical description and experimental observation of the modulation instability process and related rogue breathers in the case of stationary periodic background waves, namely cnoidal and dnoidal envelopes. Despite being well-known solutions of the nonlinear Schrodinger equation, the stability of such background waves has remained unexplored experimentally until now, unlike the fundamental plane wave. By means of two experimental setups, namely, in nonlinear optics and hydrodynamics, we report on quantitative measurements of spontaneous modulation instability gain seeded by input random noise, as well as the formation of rogue breather solutions induced by a coherent perturbation. Our results confirm the generalization of modulation instability when more complex background waves are involved.

Introduction.During the last decades, the modulation instability (MI) phenomenon have attracted a significant research interest in a variety of nearly-conservative wave systems described by the nonlinear Schrödinger equation (NLSE) in its many forms [1][2][3][4][5][6][7][8].This includes the linear stability analysis of the plane wave solution and the subsequent nonlinear stage of MI, namely the formation of localized waves such as solitons and breathers, as well as multi-breather complexes.However, beyond the plane wave solution, within the class of stationary solutions of the focusing NLSE, a wide range of periodic solutions known as cnoidal (cn) and dnoidal (dn) waves are also modulationaly unstable against small perturbations.Note that the plane wave is just a limiting case of dn-periodic waves.These solutions are highly relevant in the studies of extreme wave formation and their generalization, resulting from MI in more practical wave conditions [9][10] and from the development of integrable turbulence [11].
Although several groups have carried out mathematical investigations on the stability of such periodic waves with respect to perturbations [12][13][14][15], only a few have recently described the nonlinear stage of instability [9][10][16][17].No experiment has been reported so far.Note that propagation of such type of exact and stationary periodic envelopes has been conducted in distinct water wave facilities [18][19], but without reporting on the stability against small perturbations.For nonlinear optical studies, we can only mention the experimental evidence of the cnoidal wave self-compression in a photorefractive crystal [20].By contrast, cnoidal waves have been widely studied in the framework of the Korteweg-de Vries equation and related physical systems [2][3].
In this work, we provide an overview of both the noisedriven and the coherent seeding regimes of MI for stationary periodic waves of the NLSE.To investigate such regimes on a relevant range of parameters, an original multidisciplinary experimental approach has been required.Two complementary experimental setups based on light-wave propagation in an optical fiber and a waterwave tank are used, so that completely different timescales of the nonlinear dynamics are involved but described by the same theoretical foundation.We quantitatively confirm theoretical predictions of spontaneous MI gain as well as formation of seeded rogue wave solutions on a periodic background.Moreover, we show that the family of cnwaves is more robust against noise than dn-waves, thus being of potential interest for optical data processing and transmission [21].
Theoretical description.Our theoretical framework is based on the dimensionless form of the universal selffocusing 1D-NLSE [6], where subscripts stand for partial differentiations.Here  is a wave envelope which is a function of  (a scaled propagation distance) and  (a co-moving time with the wave-group velocity).This conventional form of the NLSE is known to be integrable and can be solved using various techniques.It has exact complex breather-type solutions as well as simpler (-) stationary (-) periodic solutions of cnoidal and dnoidal type, expressed in terms of elliptic functions [22].
The positive-definite dn-periodic waves and the signindefinite cn-periodic waves are respectively given by (2) and where  is the modulus of elliptic functions (0 ≤  ≤ 1), which gives the period of the wave function   .One can then obtain the angular frequency interval of the corresponding comb spectra   = 2/  (see illustrations in Fig. 1).For  → 1, both periodic wave families converge to the envelope soliton (sech-shape).For  → 0, these families tend to the plane wave solution with either finite-or zerovalue amplitude, respectively.From Fig. 1, it is worth noting that cn-waves are characterized by a spectral envelope mainly driven by bichromatic waves, whereas dnwaves correspond to modulated single-frequency backgrounds [19].Recall that the above waves belong to the restricted group of stationary periodic waves with trivial phase.More general elliptic wave solutions with nontrivial phase can be also analysed [15,17].
The interaction between dispersive and nonlinear effects leads to MI phenomenon for the plane wave in the presence of noise (spontaneous regime) or a weak frequency-shifted signal wave (induced regime).The linear stability analysis of periodic waves was also performed in detail (see for instance Ref. [15]).It was found that both dn-and cnwaves are modulationaly unstable with respect to longwave perturbations.We provide below the forms of MI growth rate according to parameters of the periodic waves.
The general evolution of an initial perturbation onto   or   can be expressed as   , where the MI growth rate (in amplitude) is mainly given by the real part of  = ±2√(), where () can be calculated by using the following relations for dn-and cn-waves:   =  .Here,  is the spectral parameter defined in the Lax spectrum of the Zakharov-Shabat spectral problem from Eq. (1) [10,17].The numerical scheme of computing the eigenvalues is based on the discretization of the frequency comb interval [17].We relate eigenvalues in the Lax spectrum to parameters of the periodic waves and the frequency range of perturbations that can be investigated in each frequency interval (0 ≤ || ≤   ).The instability arises only if  belongs to the bands of the Lax spectrum with {} ≠ 0. Figure 2 shows calculations of MI growth rate for both dn-and cn-periodic waves as a function of their governing parameter (i.e., the modulus of elliptic functions).In the case of dn-waves evolving from  = 0 towards 1 (see Fig. 2a), the MI starts from the plane wave limit, where it occurs for frequencies 0 ≤ |Ω| ≤ 2 (Ω c = 2) as well as characterized by a maximum growth rate equal to 1 at |Ω| = √2.Then, when  increases, the MI spectral bandwidth (here equivalent to |Ω c |) continuously reduces as the wave period T c increases.At the same time, {} also decreases and vanishes in the limit  → 1 of the stable solitons.On the contrary, for cn-waves evolving from  = 0 (i.e., the zero background limit: no MI) towards 1 (see Fig. 1b), the MI starts to grow near |Ω| = Ω c = 1 and then MI bands enlarge until reaching maximal growth rate and bandwidth for ~0.83.This maximum observed at ~0.36 Ω c remains significantly smaller than the growth rates obtained for dnperiodic waves.After that, MI growth rate decreases for higher values of  since the wave period T c increases and cn-waves tend to the stable sech-solution ( = 1).Unlike dn-waves, an original specificity of the MI phenomenon in the case of cn-waves is that the imaginary part of  is nonzero (see Figs. 2c-2d), thus leading to oscillations of the genuine growth rate along propagation distance.It means that the fastest growing perturbation changes according to {}.As a consequence, the MI growth rate indicated in Fig. 2(b) does not report the fine ξ-dependent spectral structure, but only the asymptotic solution for very large distances.
In both cases of periodic waves, the eigenvalues  ± that delimit the MI spectral bands (i.e., the end points for which {} = 0) are used to construct the rogue breather (or rogue wave, RW) solutions   on the corresponding periodic background, by using the one-fold Darboux transformation [10,17].Corresponding analytic expressions based on the above notation are given in the supplementary information.Such solutions generalize the well-known Peregrine's RW on the plane wave background.
Experimental Results.Our experimental setups are based on the propagation of arbitrarily shaped light waves in optical fibers and a common water-wave tank (detailed description in the supplementary information).Note that all measurements of MI gain and RW solutions cannot be done in both nonlinear optics and hydrodynamics due to their restricted ranges of parameters, since all the experimental parameters are embedded into the single governing parameter , thus requiring a complementary approach (detailed discussion in the supplementary information).
We first performed experimental measurements of the spontaneous MI gain when periodic waves propagate in km-long optical fibers.Here two distinct 60-and 40-GHz frequency combs are initially generated to subsequently shape exact periodic solutions (respectively for dn-and cnwaves) according to usual values of fiber dispersion  2 (-21 ps 2 km -1 ) and nonlinearity  (1.2 W -1 km -1 ), and an average wave power  0 suitably chosen and injected.The fiber loss  is very low about 4% per kilometer (in power).The correspondence between theory and experiment is the following: dimensional distance  (m) and time  (s) are given by  =   and  =  0 , where the characteristic length and time scales are   = ( 0 ) −1 and  0 = (| 2 |  ) 1/2 , respectively.The dimensional optical field  (W 1/2 ) is  =  0 1/2 .The limited resolution of the spectrum analyzer however prevents from measuring accurately the full MI bands in the vicinity of the comb peaks.This explains why the accumulated MI gain was obtained over a limited range of frequency detuning in Fig. 3(b).Even so these results are in good agreement with theoretical predictions of MI gain (the accumulated power gain over a normalized distance Δξ is obtained as 20 log 10 [ {}Δξ sin({}Δξ)]).This confirms that we almost reach a 20-dB maximal gain after 5 km of fiber at a frequency detuning of two-thirds of the comb frequency interval.The corresponding normalized distances for the theory were calculated from effective fiber lengths (this includes a correction of fiber losses to the propagation distance through   = [1 −  −  ]  ⁄ [4]).We can note the effect of fiber losses on the power of each comb harmonic after a few kilometers in Fig. 3(a).This effect is typically accompanied by a strong decrease of the modulation contrast of the dn-wave in the time domain, and even some phase-shift pulsations [19].For the case of a cn-periodic wave, the results are depicted in Fig. 3(c).We investigated the evolution of the power spectrum over 8 km of propagation and no clear signature of MI gain was observed, except a few-dB gain around the center frequency (i.e., zero-detuning).Different values of the modulus  were studied with similar results.The apparent robustness of cn-waves needs to be moderated since the studied normalized distance is only 1.63   after correction of fiber losses.In addition, two main issues could be raised based on Fig. 3(c), namely the limited resolution of the spectrum analyzer and the impact of fiber losses again observed on the power of each comb harmonic.But, in any case, Fig. 3(d) confirms that MI gain bands would be observable only if longer normalized distances are considered beyond 3   (i.e., a 12.5-km-long fiber without any propagation loss).
We underline here that the predictions are calculated by taking into account the nonzero imaginary part of , thus describing oscillations of the MI gain with distance.More specifically, it traduces the frequency location for the fastest growing perturbation after a certain distance with a maximum defined by {}.We clearly show that the accumulated MI gain differs in bandwidth and shapes for distinct propagation distances.In the first steps of propagation, a first spectral band emerges and then continuously drifts to larger frequency detunings.By increasing the propagation distance, extra sub-bands appear and fill the frequency interval defined by {} (dotted lines) while the overall gain also increases.Such complex behaviours were confirmed by numerical simulations of NLSE (see supplementary material), but their direct observation appears as a hard task from the experimental point of view.
In addition to the spontaneous MI, we carried out specific experiments in both optics and hydrodynamics about the coherent seeding of the process, and more particularly in the limiting case of end points of MI spectral bands.This corresponds to the generation of rogue breathers (RW solutions) on stationary periodic backgrounds.To this end, we use their exact solutions (see supplementary material) to shape the input periodic wave with the correct localized perturbation.According to the maximal propagation distance that can be reached, we chose suitable initial conditions ( value) to observe the maximum amplification.
Figure 4(a,c) presents both temporal and spectral evolutions measured for a rogue dn-periodic wave solution with the fiber-based light-wave platform.Our light shaping technique implies the time-periodic generation of localized perturbations, so that a 13-GHz frequency comb was initially generated to subsequently shape    (,  = −2.3)at fiber input.Note that the frequency interval for the dn-periodic wave is 78 GHz.From Fig. 4(a) we clearly reveal that the localized perturbation (centered at  = 0) grows as predicted by the theory shown in Fig. 4(b).A typical X-wave interaction forms in the space-time plane.The optical RW reaches a maximum amplitude nearly W 1/2 after 1.4 km, which is close to the theoretical prediction  0 1/2 (2 + √1 −  2 ) despite the fiber losses.As next, we also observe the RW decay just before 2 km.As expected, the nonlinear focusing of perturbation induces a significant spectral broadening shown in Fig. 4(c), and satisfies the corresponding theory from Fig. 4(d).
The range of parameters for RW solutions that can be studied with the optical platform is rather limited so that the next experiments were performed by means of the waterwave tank.We present results for both dn-and cn-periodic waves and distinct values of modulus .We recall that the surface elevation (, ) is related to the NLSE wave envelope (, ) to second-order in steepness (, ) = {(, ) (−) + 1 2 ⁄  2   2  2(−) }.The correspondence between theory and experiment can be retrieved here by using the following relations  =  2  3  and  = ( −    ⁄ )/√2, where  and  are the initial amplitude and the wave number of the carrier wave, respectively.These two parameters define the steepness , whereas the dispersion relation of linear deep-water wave theory gives the angular frequency  = () 1/2 , where  is the gravitational acceleration.The group velocity is equal to   =  2 ⁄ .Moreover, the attenuation rate in our waterwave experiments was estimated about 0.25% per meter (in amplitude), which means that the experienced dissipation (see also [23]) for RW generation will be larger here than in the optical experiment reported in Fig. 4.
Figure 5 shows the results of experiments by shaping an initial localized perturbation centered at  =0 onto dn-and cn-periodic waves (as expressed in the supplementary material) for  = -2.6.The two first cases (Fig. 5a-d) report the longitudinal evolution of perturbation for dn-periodic waves when  = 0.8 and 0.99 (i.e., close to the soliton limit) until reaching the maximal amplification after 16.8 m.In both cases, the measurements agree well with theory.For  = 0.8, the overall picture is very similar to the one reported in optics (see Fig. 4(a)), while for  = 0.99 the periodic background wave is weaker and the dynamics clearly approaches a two-soliton interaction with a maximal magnification factor near 2 (see supplementary information).
Finally, the two last cases shown in Fig. 5(e-h) look into the situation of cn-periodic waves not yet addressed.Again, the experimental results are in accordance with the theory.For  = 0.95, we retrieve similar dynamics as just previously mentioned since we are close to the soliton limit for cn-waves.Now when changing  to 0.75, one can notice the significant decrease of the nonlinear focusing experienced by the perturbation, and more particularly we confirm that maximal amplitude of the RW structure is directly proportional to the value of  according to the theory (see supplementary information).Additional experiments with distinct values of  confirmed that the amplification factor is always close to 2 (independent of ).

Conclusion.
In summary, we reported the theoretical description and direct observation of the MI process and related rogue breathers on stationary periodic cnoidal and dnoidal envelopes.The present work was performed in two distinct disciplines of wave physics, namely, optics and hydrodynamics, in order to confirm the existence of the MI phenomenon for more complex background waves than the common plane wave.We provided an overview of the main characteristics of MI gain and RWs for various values of the modulus of elliptic functions.
Future experimental research should for instance tackle the complexity of MI gain for cn-periodic waves and the instability of other wave solutions such as the double-periodic solutions [23][24][25].We expect that our multidisciplinary approach will motivate new scientific technological advances in the field of nonlinear physics, telecommunications and marine engineering.

Experimental setups
The observation of spontaneous MI gain and RW solutions cannot be done in both nonlinear optics and hydrodynamics due to their restricted ranges of parameters, since all the experimental parameters (e.g., wave period and amplitude, fiber dispersion and nonlinearity) are embedded into a single fundamental parameter, namely the modulus  of elliptic functions.For instance, the spontaneous MI grows from small random noise requires a long propagation length with almost no dissipation.Nonlinear fiber optics represents a suitable solution for this issue because the expected MI gain is lower than the plane wave limit, and spectral characterization with high dynamic range is also required.By contrast, both rogue dn-and cnperiodic waves can be observed in the water wave tank, whereas only the rogue dn-periodic wave can be generated with light waves since their exact arbitrary waveform shaping at various periodicities is far more difficult.We emphasize that all our experiments are designed in such a way as to prevent as much as possible any contribution from higher-order effects beyond the standard focusing NLSE, but losses can still affect our results.Our experimental setups (depicted in Fig. S3) are based on the propagation of arbitrarily shaped light waves in optical fibers and a common water-wave tank.Each system is capable of synthesizing nontrivial exact periodic wave profiles in the temporal domain, i.e., a prerequisite for confirming the existence of their genuine instability.
For light waves, the initial state is obtained through Fourier-transform optical pulse shaping with phase and amplitude controls in the spectral domains.This specific processing of a home-made optical frequency comb source allows to generate exact wave profiles with a specific period fixed by the frequency spacing of the optical comb.Nonlinear propagation is then studied in different lengths of the same standard single-mode fiber (SMF28) by an appropriate choice of the input average power.At fiber output, the power profiles are characterized in both time and frequency domains by means of an ultrafast optical sampling oscilloscope and an optical spectrum analyzer.In water-wave experiments conducted in deep-water conditions, the initial periodic wave profiles are shaped with a piston wave generator located at one end of the tank.An electric signal drives the piston to directly modulate the surface height in the time domain as a function of the exact mathematical expression used.The tank dimensions are 30 × 1 × 1 m 3 and the water depth is 0.7 m.A wave-absorbing beach is installed at the opposite end to avoid the influence of reflected waves.Seven wave gauges are then placed at distinct distances from the wave excitation to record the evolution of surface elevation.

FIG. 2 .
FIG. 2. Calculated MI growth rate {} as a function of normalized angular frequencies of perturbation for (a) dn-waves and (b) cn-waves.(c,d) Corresponding calculated {}.The plane-wave limit (PW) is plotted with a black line in panel (a).

Figure 3 (
a-b) shows corresponding recorded power spectra | ̃|2 and accumulated MI gain obtained for various propagation distances in the case of a dn-periodic wave ( = 0.7).We clearly observe the MI gain bands emerging around the central peak of the dn-wave in Fig. 3(a) and growing exponentially with propagation distance.Their bandwidth remains limited by the intrinsic frequency spacing of the comb formed by the dn-wave (i.e., 60 GHz).

FIG. 3 .
FIG. 3. (a) Experimental power spectra of spontaneous MI for a dn-wave ( = 0.7) on various propagation distances (here  0 = 0.6 W).(b) Accumulated MI power gain deduced from (a) and compared to theory (black dotted lines) calculated from respective normalized distances 0.7, 1.38, 2.03, 2.65 and 3.25   based on the effective fiber lengths.(c) Experimental power spectra of spontaneous MI for a cn-wave ( = 0.92) on various distances (here  0 = 0.2 W).No MI gain is clearly apparent as the corresponding normalized distances are respectively 0.46, 0.88, 1.27, and 1.63   based on the effective fiber lengths.(d) Theoretical predictions of accumulated MI power gain, for the cnwave studied in (c), when longer distances are considered.Solid (dotted) lines are calculated with (without) {} .