Direct scattering transform: catch soliton if you can

Direct scattering transform: catc h soliton if y ou can Andrey Gelash 1 , 2 ∗ and Rustam Mully adzhanov 3 , 4 † 1 Institute of Automation and Ele ctr ometry SB RAS, Novosibirsk 630090, Russia 2 Skolkovo Institute of Scienc e and T e chnolo gy, Mosc ow 121205, R ussia 3 Institute of Thermophysics SB RAS, Novosibirsk 630090, Russia and 4 Novosibirsk State University, Novosibirsk 630090, Russia Direct scattering transform of nonlinear w av e fields with solitons ma y lead to anomalous numer- ical errors of soliton phase and position parameters. With the fo cusing one-dimensional nonlinear Sc hr¨ odinger equation serving as a model, w e in v estigate this fundamen tal issue theoretically . Using the dressing metho d we find the landscap e of soliton scattering co efficien ts in the plane of the com- plex spectral parameter for multi-soliton wa v e fields truncated within a finite domain, allowing us to capture the nature of particular numerical errors. They dep end on the size of the computational domain L leading to a coun terintuitiv e exp onen tial divergence when increasing L in the presence of a small uncertaint y in soliton eigen v alues. In contrast to classical textbo oks, we reveal how one of the scattering coefficients loses its analytical properties due to the lack of the wa ve field com- pact supp ort in case of L → ∞ . Finally , we demonstrate that despite this inherit direct scattering transform feature, the wa ve fields of arbitrary complexity can be reliably analysed. Intr o duction . – Since the 1970s we hav e observ ed an impressiv e progress in nonlinear mathematical ph ysics stim ulated b y the disco v ery of the complete in tegrabil- it y of some nonlinear partial differential equations [1 – 4]. Among them are the one-dimensional Korteweg–de V ries (KdV) [5] and nonlinear Sc hr¨ odinger (NLSE) [6] equa- tions serving as the fundamental nonlinear wa ve mo dels and app earing in different areas of physics. This break- trough has taken place due to the developmen t of the inverse sc attering tr ansform (IST) [1, 2] allowing one to solv e the initial-v alue problem in terms of the nonlin- ear harmonics decomp osition representing the sc attering data or the IST sp e ctrum . The spectrum can b e found using the dir e ct sc attering tr ansform (DST) leading to the full kno wledge of the nonlinear w av e field evolution go verned by the in tegrable differen tial equation [1, 2]. After sev eral decades of analytical studies of in tegrable equations, the rapid growth of interest to describ e ar- bitrary shap ed, noisy and even random nonlinear wa ve fields has promoted the need in accurate n umerical meth- o ds for the DST. The Boffetta–Osb orne metho d [7] rep- resen ts the first n umerical realization of the DST follo wed b y a sequence of further improv ements and alternativ es [8 – 12]. These adv ancements ha ve made the DST an es- sen tial scien tific to ol with a wide range of theoretical and exp erimen tal applications [4, 13–17]. The remark- able abilit y of the DST to identify and characterise soli- tons representing the coheren t structures in nonlinearly in teracting wa v e fields provides fundamen tal information ab out the origin of v arious ph ysical effects [18, 19] and can b e fruitfully used in practical applications such as optical telecommunication systems [20 – 23]. A significant amoun t of work has b een devoted to understand the dis- tribution of soliton amplitudes and v elo cities (eigen v al- ues) [24, 25] with particular fo cus on its role in propaga- tion of o cean w av es [16, 21, 26] and optical pulses [18, 27]. Although the soliton eigenv alues can be found using man y v ariations of the n umerical DST [28, 29], an accu- rate iden tification of soliton phase and p osition parame- ters represented b y the so-called norming c onstants is still a challenging problem. So far the existing approac hes for finding b oth eigenv alues and norming constan ts hav e demonstrated success only for a relativ ely simple w a ve fields con taining up to fiv e solitons [7, 12, 29]. The recent dev elopment of high-order n umerical sc hemes has allo wed to process large multi-soliton wa ve fields with particular examples of 128 solitons, revealing sev eral types of nu- merical instabilities including the non-trivial b eha viour of norming constan ts [30]. The first class of the insta- bilities is a result of accum ulation of discretization er- rors during the scattering through a large wa v e field, whic h can be efficiently resolved b y high-order schemes [30]. The second class is related to round-off errors dur- ing the computation of the scattering co efficien ts, whic h can b e fixed by high-precision arithmetics similar to the IST pro cedure [19]. The third class represents anoma- lous errors for the norming constants, when the eigen v al- ues are computed without sufficien t accuracy , requiring high-precision arithmetics for eigenv alue identification. In this work w e theoretically reveal the nature of these anomalous errors within the DST framew ork making soli- ton phase and p osition characteristics extremely elusive. As a mo del w e consider the fo cusing NLSE for a complex w av e field ψ ( t, x ), which in the non-dimensional form is as follows: iψ t + 1 2 ψ xx + | ψ | 2 ψ = 0 , (1) where t and x are the time and spatial co ordinate. The soliton parameters represent the discrete part of the scat- tering data, which can b e found via the so-called sc at- tering c o efficients . Instead of standard scattering co effi- cien ts defined on an infinite line, w e use their “truncated” analogues on a finite in terv al leading to the sensible the- oretical analysis of the error sources within the DST. In 2 particular, employing the dressing method, w e find the landscap e of the scattering co efficien ts in the complex plane of the sp ectral parameter for a multi-soliton w av e field and track the b eha viour of nu merical errors for dif- feren t configurations of scattering data leading to specific recip es on e rr or reduction. W e sho w that these errors for norming constan ts de- p end on the size of the computational domain L lead- ing to a counterin tuitive exponential div ergence when in- creasing L in the presence of a small uncertain ty in soli- ton eigen v alues. In contrast to classical textb ooks , w e study how one of the scattering co efficien ts loses its an- alytical prop erties due to the lack of the wa v e field com- pact supp ort in case of L → ∞ . Finally , we demonstrate that despite this inherit DST feature, the wa ve fields of arbitrary complexity can b e reliably analysed. The dir e ct sc attering tr ansform for the NLSE . – The IST theory establishes a link b et ween the focusing NLSE, see Eq. (1), and the follo wing auxiliary Zakharo v–Shabat (ZS) linear system for the t w o-comp onen t vector wa v e function Φ ( t, x, ζ ) = ( φ 1 , φ 2 ) [6]: Φ x = b Q ( ψ ) Φ , b Q =  − iζ ψ − ψ ∗ iζ  , (2) where ζ = ξ + iη is a sp ectral parameter, the star de- notes the complex-conjugate v alue. The ZS system fea- tures a sp ectrum comp osed of con tinuous and discrete parts with the latter located on the complex plane. As t ypically done, we consider the upp er half of the com- plex plane with η > 0 since ζ ∗ = ξ − iη correspond to the same class of NLSE solutions. The contin uous sp ec- trum o ccupies only the real axis ξ ∈ R , while the discrete part (eigen v alues) is represen ted by the complex p oin ts with its total num b er equal to N . The scattering data of the p oten tial ψ ( t, x ) is traditionally in tro duced using a solution of the ZS system with ζ = ξ and the following asymptotics at infinity: lim x →−∞ Φ =  e − iξx 0  , lim x →∞ Φ =  a ( ξ ) e − iξx b ( ξ ) e iξx  , (3) where a ( ξ ) and b ( ξ ) are the scattering co efficien ts. The first co efficien t has an analytic contin uation a ( ζ ) to the ζ -plane with zeros at the discrete eigenv alues ζ k with k = 1 , ..., N . The second co efficien t b ( ξ ) is defined on the real axis and at the eigenv alue p oin ts ζ k with b ( ζ k ) = b k . It is important to emphasize that b ( ξ ) can b e analytically con tin ued to the ζ -plane only when the p oten tial ψ ( x ) has c omp act supp ort , i.e. if ψ = 0 outside of a compact set on the x -line [31]. As mentioned earlier, the total scattering data represen ts the discrete { ζ k , ρ k } and contin uous { r } sp ectrum: a ( ζ k ) = 0 , ρ k = b k a 0 ( ζ )     ζ = ζ k ; r ( ξ ) = b ( ξ ) a ( ξ ) , (4) where ρ k is the complex-v alued norming constants asso- ciated with ζ k and r ( ξ ) is the reflection co efficien t. Eac h Figure 1: T ypical multi-soliton wa ve field used for demon- stration of the DST numerical errors with N = 6. The 6- S S solution is obtained using Eqs. (5) and (6). See Fig. 2 on the sp ectral con tent of this solution. discrete eigen v alue ζ k = ξ k + iη k corresp onds to a soli- ton in the wa ve field with the amplitude A k = 2 η k and group v elocity V k = 2 ξ k , while r ( ξ ) describes nonlinear disp ersiv e wa v es. In case of r ( ξ ) = 0, i.e. the disp ersiv e w a ves are absent, the w av e field corresp onds to N -soliton solution ψ N S S ( x ) whic h can be reconstructed analytically employing the scattering data (4) [1] (see also [30]): ψ N S S ( x ) = − 2 iρ k e iζ k x [( E + M ∗ M ) − 1 ] k,j e iζ j x . (5) Here E is the N × N unity matrix and the elements: M k,j = iρ j ( ζ ∗ k − ζ j ) − 1 e − i ( ζ ∗ k − ζ j ) x . (6) The norming constant can be con venien tly parametrized as follows: ρ k = − iA k e A k x 0 k − iθ k , (7) where tw o real-v alued parameters x 0 k and θ k describ e the p osition in space and phase of the corresp onding soliton [32]. Below we use an example shown in Fig. 1 demon- strating a typical multi-soliton w a v epack et with N = 6 (6- S S ) constructed with the help of the abov e form u- las with a random distribution of { θ k } and c hosen set of { x 0 k } . F or ψ N S S ( x ) the first scattering co efficien t is known in a closed form in the ζ -plane [1]: a N ( ζ ) = N Y k =1 ζ − ζ k ζ − ζ ∗ k , (8) while the second scattering coefficient b N ( ξ ) cannot be analytically con tinued to ζ -plane due to infinite exp o- nen tially decaying tails of ψ N S S ( x ) contradicting to the necessary compact supp ort prop ert y . T runc ation of the wave field . – W e b egin our theo- retical analysis of the DST introducing a finite domain 3 [ − L, L ], where the wa v e field ψ ( x ) is well lo calized. The truncation of ψ guarantees the compact support allowing to define b tr ( ζ ) in the complex plane, where the subscript ‘ tr ’ highlights that the corresp onding ψ is non-zero only inside the domain [ − L, L ]. W e introduce a wa ve function Φ tr with shifted b oundary conditions from x → ±∞ , see Eq. (3), to x = ± L : Φ tr ( − L ) =  e − iζ L 0  , Φ tr ( L ) =  a tr ( ζ ) e − iζ L b tr ( ζ ) e iζ L  , (9) while with the help of a tr ( ζ ) and b tr ( ζ ) we define ρ tr ( ζ ) = b tr ( ζ ) a 0 tr ( ζ ) . (10) Our key result is the theoretical deriv ation of a N ,tr ( ζ ) and b N ,tr ( ζ ) corresp onding to N -soliton p oten tials in the ζ -plane. Using the dressing metho d to construct solu- tions of ZS problem for ψ N S S and assuming large enough L , w e obtain the follo wing expressions (see Supplemen- tary materials for a detailed deriv ation): a N ,tr ( ζ ) = a N ( ζ ) + o N , (11) b N ,tr ( ζ ) = a N ( ζ ) N X k =1 ρ k e − 2 i ( ζ − ζ k ) L ζ − ζ k + o N . (12) It is conv enien t to extract a N ( ζ ) in Eq. (12), although it cancels out the denominator ( ζ − ζ k ) − 1 for all k when the full expression (8) is explicitly employ ed. A new notation is used to shorten the presentation: o N = p ( ζ ) o ( e − 2 η min L ) , (13) whic h is based on a “little- o ” [33], a rational function p ( ζ ) and the expression η min = min[ η 1 , .., η N ] represent- ing the minimum v alue among the considered set of η . Fig. 2 sho ws the t ypical behaviour of a N ,tr ( ζ ) around the eigen v alues and stiff exp onen tial growth of b N ,tr ( ζ ) for the 6- S S solution presented in Fig. 1. W e should stress that b oth form ulas (11) and (12) are v erified nu- merically in the ζ -plane, see Supplemen tary materials. Ho wev er, concerning the accuracy of b N ,tr close to the real axis a small deviation can still arise due to the fact that the leading order term and o N b ecome of the same order. This region of in terest has recently b een addressed [34]. The expression (12) cov ers a fundamen tal issue on the analytical prop erties of b N ( ζ ) in the complex plane lead- ing to the following result: b N ( ζ ) = lim L →∞ b N ,tr ( ζ ) =          b k , ζ = ζ k , ∞ , η > η min ; ζ 6 = ζ k , b min ( ζ ) , η = η min , 0 , η < η min , (14) where b min ( ζ ) = a N ( ζ ) ρ min e − 2 i ( ζ − ζ min ) L / ( ζ − ζ min ) with the subscript ‘ min ’ corresp onding to the soliton with the Figure 2: Behaviour of the scattering co efficien ts in ζ -plane according to Eqns. (11) and (12) with L = 50. Green dots sho w { ζ k } corresponding to the wa ve pack et demonstrated in Fig. 1. minimal η as defined ab o v e. Th us, the second scattering co efficien t b N ( ζ ) is analytic only inside the band 0 ≤ η < η min . Anomal ous err ors . – Ideally , when L tends to infin- it y , one exp ects to end up with the exact formulation of the problem, see Eq. (3). Ho wev er, one of the counter- in tuitive results of the present work boils down to the fact that increasing L leads to a n um b er of numerical difficulties when trying to determine norming constants. Indeed, expressions (10) and (12) sho w that at large L ev en a small deviation δζ k of the corresp onding computed eigen v alue ζ k can lead to large errors for a norming con- stan t ρ k . In order to sho w that, w e expand the second scattering co efficien t (12) in the vicinity of ζ k : b N ,tr ( ζ k + δ ζ k ) ≈ ρ k ζ k − ζ ∗ k N Y j 6 = k ζ k − ζ j ζ k − ζ ∗ j | {z } I term + (15) δ ζ k  N X l 6 = k ρ l e − 2 i ( ζ k − ζ l ) L ( ζ k − ζ l )( ζ k − ζ ∗ k ) N Y j 6 = k ζ k − ζ j ζ k − ζ ∗ j  | {z } II term . According to the definition (10), the deviation in the norming co efficient ρ N ,tr ( ζ k + δ ζ k ) ma y b e exp onen tially large, caused by the second term in (15) for b N ,tr , while a 0 N ,tr ( ζ k + δ ζ k ) do es not hav e this problem. F or eac h soli- ton the v alue of the error, b eing a function of L , can b e estimated using the largest exp onen t of the term I I, see (15): error[ ρ k ]( L ) ∼ term I I ∼ e 2( η k − η min ) L . (16) The error b ecomes critical with the increase of L when b oth terms are of the same order in the expression (15): term I ∼ term I I . (17) 4 T o get a feeling on the order of the deviation δ ζ leading to the condition (17), one can insp ect the results for a simple tw o-soliton case: δ ζ cr 1 ∼ ρ 1 ρ 2 ( ζ 1 − ζ 2 ) e − 2 i ( ξ 2 − ξ 1 ) L e 2( η 2 − η 1 ) L , (18) δ ζ cr 2 ∼ ρ 2 ρ 1 ( ζ 2 − ζ 1 ) e − 2 i ( ξ 1 − ξ 2 ) L e 2( η 1 − η 2 ) L , (19) where the subscript ‘ cr ’ denotes the critical v alue. As- suming η 2 > η 1 without los s of generalit y , we obtain an exp onen tial divergence of δ ζ cr 1 with L , while δ ζ cr 2 on the con trary tends to zero. This fact means that in order to reduce the error when computing b N ,tr , one has to guar- an tee that the eigen v alue is computed with the appropri- ate accuracy , being demanding for δ ζ cr 2 → 0. The par- ticular num b er of necessary digits can b e estimated from the eigenv alue difference and the v alue of L , although it is ob vious that the required p recision in ma jorit y of cases is more demanding than 10 − 16 corresp onding to the stan- dard machine precision. At the same time δ ζ cr 1 → ∞ as L → ∞ suggesting that the soliton with the small- est eigen v alue does not suffer from this instability . Note that δ ζ cr 1 , 2 in (18), (19) also contain the ratio of norming constan ts which exp onen tially dep end on the soliton p o- sitions x 0 1 , 2 , see the parametrization (7), whic h can also b e an additional stiff condition. This picture stays the same when more solitons in the w a v epack et are consid- ered as is shown b elo w with a n umerical example. Numeric al discr etization and examples . – A practical implemen tation of the DST implies the influence of the n umerical discretization on the obtained analytic results. The eigenv alue condition a ( ζ k ) = 0, see Eq. (4), trans- forms into: a num ( ζ num k ) = 0 , (20) where the sup erscript ‘ num ’ denotes the numerical (dis- cretized) counterpart of the initial problem. Note that this kind of errors should not b e confused with ones caused by the domain truncation. As w as shown for N - soliton p oten tials, the relation a N ,tr = a N is accurate within exp onen tially small terms o N for large L , see Eq. (11), meaning that the discrete sp ectrum { ζ k } do es not c hange within o N after truncation. W e pro vide n umerical results proving this fact, see Supplemen tary materials. W e argue that expressions for scattering coefficients (11) and (12) for the truncated ψ N S S ( x ) preserv e the same structure after discretization, although exact v alues { ζ k } sligh tly shift to { ζ num k } . This hypothesis is strongly supp orted by the follo wing n umerical results presen ted b elo w and Supplemen tary materials. In the end w e study the features of b num N ,tr sho wing that the numerical results are in go o d agreemen t with expression (12), although additional sources of errors are inv olved but turn out to b e irrelev ant. W e p erform the numerical DST of 6- S S solution pre- sen ted ab o ve as an example, see Figs. 1 and 2, v arying Figure 3: Influence of the size of the numerical domain on the errors of { ζ num k } and { ρ num k } for 6- S S presented abov e, see Figs. 1, 2. (a) Absolute errors for soliton eigenv alues. (b) Relativ e errors for soliton norming constants computed using { ζ num k } obtained with a standard machine precision and (c) high-precision arithmetics. In the bottom w e sho w a sc hemat- ics for the right ( x > 0, black) and left ( x < 0, blue) part of the wa ve field as in Fig. 1. the width of the domain [ − L, L ]. Fig. 3 demonstrates the influence of L on the errors of the computed discrete sp ec- trum { ζ num k , ρ num k } compared to exact v alues. F or that w e solv ed the ZS system (2) using the standard second- order accuracy Boffetta–Osb orne metho d [7] keeping the discretization step constan t for all the cases. In Sup- plemen tary materials we v erify that high-order metho ds [30] demonstrate similar results. When L is large enough, { ζ num k } can alw a ys be reliably identified with the accu- racy corresp onding to the chosen numerical scheme and the w a ve field discretization as illustrated in Fig. 3(a). Note that for a large particular eigenv alue ζ k the accuracy | ζ num k − ζ k | is low er than for small-amplitude solitons. As exp ected, the reduction of the domain makes the eigen- v alues undetectable for solitons exp osed to truncation, 5 whic h is easy to observe in Fig. 3(a) in comparison with the schematic w a ve field profile demonstrated in the b ot- tom of Fig. 3(c). The main result of the work is presen ted in Figs. 3(b,c) where the influence of L on the errors of the calcu- lated norming constan ts { ρ num k } is considered. Firstly , w e demonstrate the results of calculations when { ζ num k } are computed from the condition (20) with the standard double (machine) precision leading to δ ζ k ∼ 10 − 16 in the expression (15). Fig. 3(b) sho ws that for large enough L these deviations in eigenv alues lead to the exp onen tial gro wth of errors as predicted b y the dev elop ed theory , see Eq. (16). A t the same time, for smaller L < 15, as ex- p ected, we observe the truncation errors for the norming constan ts. These tw o effects mak e the whole set of solitons not p ossible to identify at any fixed v alue of L , in particular the third and fourth solitons sho wn b y green and purple dots are completely “uncatchable” when standard preci- sion is used. Secondly , w e p erform the same set of simu- lations employing high-precision arithmetics to identify { ζ num k } and calculate { ρ num k } excluding the describ ed anomalous errors. Fig. 3(c) sho ws that we successfully obtain { ρ num k } when L is large enough so that the wa ve field is well lo calised inside the computational domain. Conclusions . – In this w ork we consider N -soliton solu- tions of the nonlinear Sc hr¨ odinger equation and the cor- resp onding solutions of the Zakharo v–Shabat problem. The main result is the theoretical deriv ation of the con- nection b et ween scattering co efficien ts a N and b N defined for a problem on the infinite line with a N ,tr and b N ,tr cor- resp onding to the same problem on a finite (truncated) domain of the width 2 L . Using the dressing metho d we obtain closed-form expressions for a N ,tr and b N ,tr allo w- ing us to express b N = lim b N ,tr at L → ∞ and demon- strate its analytic prop erties. Based on these results we rev eal a new class of inher- en t instabilities of the direct scattering transform lead- ing to anomalous numerical errors of norming constants gro wing exponentially with L . A high-precision arith- metic is required in order to exclude these errors lead- ing to the fact that hybrid metho ds employing tw o dif- feren t numerical approaches for a subsequent computing of eigenv alues and norming constants should b e applied with caution due to p ossible systematic errors in eigenv al- ues. Note that small v ariations of the input wa ve fields do not affect the ov erall stability of the DST. W e ex- p ect that the presence of solitons in complex w av e fields con taining contin uous sp ectrum is alwa ys manifested by rapidly changing landscap e of ρ tr ( ζ ), representing a gen- eral situation when the suggested strategy to p erform the DST can b e applied. In particular this idea is supp orted b y results for the exactly solv able rectangular p oten tial mo del revealing exp onen tial growth of the scattering co- efficien ts in the complex plane with the p oten tial length [35]. In addition, our results can b e straightforw ardly generalised to another imp ortan t class of coherent struc- tures app earing in the NLSE – breathers [36 – 40], as well as to other integrable mo dels. These insights give theo- retical foundations to develop robust algorithms for the calculation of scattering data of complex w a ve fields to study v arious nonlinear phenomena. The complete iden- tification of coheren t structures in sto c hastic wa ve fields is on top of the current agenda [17, 41 – 45]. A cknow le dgments . – Both authors (A.G. and R.M.) prop osed key ideas and con tributed equally to theoreti- cal computations, n umerical simulations and manuscript preparation. A.G. ac knowledges supp ort of RFBR gran t No. 19-31-60028, R.M. ackno wledges supp ort of RSF gran t No. 19-79-30075. Section “Anomalous errors” re- p orts the results of the w ork supported solely by RSF gran t No. 19-79-30075. The authors thank Profs. E.A. Kuznetso v, V.E. Zakharov and the group of Prof. D.A. Shapiro for fruitful discussions. ∗ Electronic address: agelash@gmail.com † Electronic address: rustammul@gmail.com [1] S. No vik ov, S. Manak ov, L. Pitaevskii, and V. Za- kharo v, The ory of solitons: the inverse sc attering metho d (Springer Science & Business Media, 1984). [2] M. J. Ablowitz and H. Segur, Solitons and the inverse sc attering tr ansform , V ol. 4 (Siam, 1981). [3] A. C. Newell, V ol. 48 (Siam, 1985). [4] A. Osborne, Nonline ar o c e an waves (Academic Press, 2010). [5] C. S. Gardner, J. M. Greene, M. D. Krusk al, and R. M. Miura, Physical Review Letters 19 , 1095 (1967). [6] V. Zakharo v and A. Shabat, Soviet Ph ysics JETP 34 , 62 (1972). [7] G. Boffetta and A. R. Osb orne, Journal of Computational Ph ysics 102 , 252 (1992). [8] S. Burtsev, R. Camassa, and I. Timofeyev, Journal of Computational Physics 147 , 166 (1998). [9] L. L. F rumin, O. V. Belai, E. V. Podivilov, and D. A. Shapiro, Journal of the Optical So ciet y of America B 32 , 290 (2015). [10] A. V asylc henko v a, J. E. Prilepsky , and S. K. T uritsyn, Optics Letters 43 , 3690 (2018). [11] S. Medv edev, I. V asev a, I. Chekhovsk oy , and M. F e- doruk, Optics Letters 44 , 2264 (2019). [12] F. J. Garca-Gmez and V. Aref, Journal of Light w av e T ec hnology 37 , 3563 (2019). [13] J. E. Prilepsky , S. A. Derevy anko, K. J. Blo w, I. Gabito v, and S. K. T uritsyn, Ph ysical Review Letters 113 , 013901 (2014). [14] I. T av akkolnia and M. Safari, Optics Express 25 , 18685 (2017). [15] S. K. T uritsyn, J. E. Prilepsky , S. T. Le, S. W ahls, L. L. F rumin, M. Kamalian, and S. A. Derevyank o, Optica 4 , 307 (2017). [16] A. Slun yaev, Radioph ysics and Quantum Electronics 61 , 1 (2018). [17] S. Randoux, P . Suret, A. Chab c houb, B. Kibler, and G. El, Physical Review E 98 , 022219 (2018). 6 [18] S. Derevyank o and E. Small, Ph ysical Review A 85 , 053816 (2012). [19] A. A. Gelash and D. S. Agafontsev, Physical Review E 98 , 042210 (2018). [20] S. K. T uritsyn and S. Derevyank o, Ph ysical Review A 78 , 063819 (2008). [21] M. I. Y ousefi and F. R. Kschisc hang, IEEE T ransactions on Information Theory 60 , 4312 (2014). [22] Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Y ousefi, C. Lu, P .-K. A. W ai, F. R. Ksc hischang, and A. P . T. Lau, IEEE Photonics T echnology Letters 27 , 1621 (2015). [23] L. L. F rumin, A. Gelash, and S. K. T uritsyn, Physical Review Letters 118 , 223901 (2017). [24] J. C. Bronski, Physica D: Nonlinear Phenomena 97 , 376 (1996). [25] M. Klaus and J. Sha w, Physical Review E 65 , 036607 (2002). [26] A. Osb orne, E. Segre, G. Boffetta, and L. Cav aleri, Phys- ical Review Letters 67 , 592 (1991). [27] F. Braud, M. Conforti, A. Cassez, A. Mussot, and A. Kudlinski, Optics Letters 41 , 1412 (2016). [28] J. Y ang, Nonline ar waves in integr able and noninte gr able systems , V ol. 16 (SIAM, 2010). [29] A. V asylchenk ov a, J. Prilepsky , D. Shep elsky , and A. Chattopadhy ay , Communications in Nonlinear Science and Numerical Simulation 68 , 347 (2019). [30] R. Mully adzhanov and A. Gelash, Optics Letters 44 , 5298 (2019). [31] L. D. F addeev and L. A. T akh ta jan, Hamiltonian metho ds in the the ory of solitons (Springer Science & Business Media, Berlin, 2007). [32] G. L. Lamb, Elements of soliton the ory (New Y ork, Wiley-In terscience, 1980). [33] G. H. Hardy , E. M. W right, et al. , An intr o duction to the the ory of numb ers (Oxford universit y press, 1979). [34] V. Aref, in SCC 2019; 12th International ITG Confer enc e on Systems, Communic ations and Co ding (VDE, 2019) pp. 1–6. [35] S. Manako v, Zh. Eksp. T eor. Fiz 65 , 10 (1973). [36] E. Kuznetso v, in A kademiia Nauk SSSR Doklady , V ol. 236 (1977) pp. 575–577. [37] Y.-C. Ma, Studies in Applied Mathematics 60 , 43 (1979). [38] D. Peregrine, J. Austral. Math. Soc. Ser. B 25 , 16 (1983). [39] N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, So v. Phys. JETP 62 , 894 (1985). [40] C. Kharif, E. Pelino vsky , and A. Sluny aev, R ogue waves in the o c ean, observation, the ories and modeling (Ad- v ances in Geoph ysical and Environmen tal Mechanics and Mathematics Series, Springer, Heidelb erg, 2009). [41] J. M. Soto-Cresp o, N. Devine, and N. Akhmediev, Ph ys- ical Review Letters 116 , 103901 (2016). [42] M. Conforti, S. Li, G. Biondini, and S. T rillo, Optics Letters 43 , 5291 (2018). [43] S. T rillo and M. Conforti, Optics Letters 44 , 4275 (2019). [44] A. E. Krayc h, D. Agafontsev, S. Randoux, and P . Suret, Ph ysical Review Letters 123 , 093902 (2019). [45] A. Gelash, D. Agafon tsev, V. Zakharov, G. El, S. Ran- doux, and P . Suret, Physical Review Letters (2019).