Doubly periodic solutions of the focusing nonlinear Schr"odinger equation: recurrence, period doubling and amplification outside the conventional modulation instability band

Doubly p erio dic solutions of the fo cusing nonlinear Sc hr¨ odinger equation: recurrence, p erio d doubling and amplification outside the con v en tional mo dulation instabilit y band Matteo Conforti, 1 ∗ Arnaud Mussot, 1 Alexandre Kudlinski, 1 Stefano T rillo 2 and Nail Akhmediev 3 1 University of Lil le, CNRS, UMR 8523PhLAMPhysique des L asers A tomes et Mol ´ ecules, Lil le, F r ance 2 Dep artment of Engine ering, University of F err ar a, F err ar a, Italy 3 Optic al Scienc es Gr oup, Dep artment of The or etic al Physics, R ese ar ch Scho ol of Physics, The Austr alian National University, Canb err a, ACT 2600, Austr alia (Dated: No vem ber 7, 2021) Solitons on a finite a bac kground, also called breathers, are solutions of the focusing nonlinear Sc hr¨ odinger equation, which play a piv otal role in the description of rogue w a ves and modulation instabilit y . The breather family includes Akhmediev breathers (AB), Kuznetso v-Ma (KM), and P eregrine solitons (PS), which ha v e b een successfully exploited to describ e sev eral physical effects. These families of solutions are actually only particular cases of a more general three-parameter class of solutions originally deriv ed b y Akhmediev, Eleonskii and Kulagin [Theor. Math. Phys. 72 , 809–818 (1987)]. Having more parameters to v ary , this significan tly wider family has a potential to describ e many more ph ysical effects of practical interest than its subsets mentioned ab ov e. The complexit y of this class of solutions preven ted researc hers to study them deeply . In this pap er, w e o v ercome this difficult y and report sev eral new effects that follow from more detailed analysis. Namely , we present the doubly p eriodic solutions and their F ourier expansions. In particular, w e outline some striking properties of these solutions. Among the new effects, we men tion (a) regular and shifted recurrence, (b) p erio d doubling and (c) amplification of small p eriodic perturbations with frequencies outside the conv en tional modulation instability gain band. P ACS n um b ers: I. INTR ODUCTION The nonlinear Sc hr¨ odinger equation (NLSE) is one of the paradigms of mo dern nonlinear science. It describ es the evolution of narrow-band env elopes under the com- bined action of weak dispersion and non-linearity , and naturally app ears in differen t branches of ph ysics such as optics [1], hydrodynamics [2], plasma [3] and cold atoms [4]. In the fo cusing regime, the contin uous w av e (CW) so- lution of NLSE is unstable with resp ect to modulation in- stabilit y (MI) [5, 6], whic h en tails the exp onen tial gro wth of lo w-frequency p erturbations. Mo dulation instability is the most basic and widespread nonlinear phenomenon, whic h has b een studied since the 60s in suc h a diverse disciplines as h ydrodynamics [5] or nonlinear optics [6]. While the initial developmen t of MI that arises from the linear stabilit y analysis is well understo o d, the nonlin- ear stage of MI is an extremely hot and active research topic. In the fully nonlinear regim e, MI can give rise to F ermi-P asta-Ulam recurrences [7–15], mo dulated cnoidal w av es [16 – 20], deterministic formation of breathers [21– 26], whose app earance can b e triggered b y the shape of the initial perturbation of the background [27, 28], as w ell as tubulent states mediated by statistical app ear- ance of breathers [29–31]. Moreov er, MI has b een iden- tified as one of the possible generating mechanisms of rogue wa v es [32 – 34]. The latter can b e describ ed analyt- ically as breathers, or solitons on finite background. All the solutions that describe suc h regimes can be presen ted in analytic form due to the in tegrability of the NLSE and the inv erse scattering transform (IST) being the p o werful to ol for its analysis. There are different form ulations of IST dep ending on sp ecific b oundary con- ditions of the problem to b e solved. F or example, prob- lems related to MI require p erio dic b oundary conditions. These set of problems is more complicated than the one with zeros at infinity . Thus, sp ecial finite-band integra- tion theory has b een developed [35 – 38]. This tec hnique p ermits one to write the solutions of the NLSE with p eri- o dic b oundary conditions explicitly as ratios of Riemann theta functions [39–43]. How ev er, these formal solutions con tain an infinite num b er of free parameters and their practical use remains questionable. Indeed, extracting ph ysically relev ant solutions from these general ones is not an easy task. Practical solutions are mostly relying on direct metho ds with the most p opular one b eing the Hirota metho d [44]. An alternative wa y to pro ceed is constructing more complex solutions from simple ones rather than trying to extract simple solutions from the most general one. One of these techniques is Darb oux transform that al- lo ws one to start with seeding solution in the form of a plane wa v e and build perio dic solutions with elab orate ev olution [45, 46]. Using the plane wa ve as a seed solu- tion only allows one to find some classes of solutions that can b e considered as higher-order MI. In order to expand this class of solutions to more general ones, other types of seed solutions must be used. The basic family of solutions of the NLSE that is peri- o dic b oth in space and time can b e found using a sp ecial ansatz suggested in [47, 48]. This technique allo ws one to 2 reduce the NLSE which is an infinite dimensional Hamil- tonian system to a finite dimensional one that can b e then solv ed analytically [45, 47–49]. The result is the family of first-order solutions that ha v e arbitrary amplitude and arbitrary p erio ds in time and space [48]. W ell kno wn so- lutions suc h as Peregrine soliton (PS) [51], Kuznetso v-Ma (KM) soliton [52, 53] or Akhmediev breather (AB) [47] and the tw o-parameter family of doubly p erio dic solu- tions of A and B-types [11, 47, 50] are sp ecial cases of this significantly wider family . Thus, we can talk ab out this three parameter family as the most general known family of doubly p erio dic solutions. In this pap er, w e rep ort a detailed study of the prop- erties of the first-order solutions, which include all ab o v e listed subsets and pro vide an analytic description for sev- eral new in triguing nonlinear phenomena. These include: (1) FPU recurrence as a natural consequence of longi- tudinal p erio dicit y , (2) p eriod doubling during the MI pro cess of growing p erio dic p erturbation and (3) the MI gro wth outside of the conv en tional MI band. F or the first time, w e found the F ourier co efficien ts of the first order solutions in analytic form. The latter is an essential part of the present work that made it p ossible the expansion of the num ber of physical applications of the doubly p e- rio dic solutions. The paper is organised as follows. In Sec. I I, w e giv e the exact form of the family of solutions with three free parameters. Moreo ver, w e present the expressions of the F ourier co efficien ts and analyse the relev an t phys- ical prop erties suc h as the amplitude, the perio d and the w av en um b er of the solutions as a function of these three gov erning parameters. In Sec. II I w e predict, based on these solutions, the new counter-in tuitive phenomena suc h as the gro wth of linearly stable p erturbations and p eriod doubling within the MI. In Sec. IV we summarise our results. I I. FIRST-ORDER DOUBL Y PERIODIC SOLUTIONS W e start from the NLSE written in the following nor- malised form, whic h is the standard nonlinear fiber optics notation: iψ z + 1 2 ψ tt + | ψ | 2 ψ = 0 . (1) The first-order solutions ha v e the following prop ert y: at each propagation step z , the real and imaginary parts of the field ψ ( t, z ) are linearly related [47]. This implies the following form of the solution [48]: ψ ( t, z ) = [ Q ( t, z ) + iδ ( z )] e iφ ( z ) , (2) where Q , δ and φ are real functions. F or every v alue of z , Eq.(2) represents a straigh t line in the complex plane ( Reψ , I mψ ). The ansatz (2) permits to reduce the NLSE, whic h is an infinite-dimensional Hamiltonian system, to a finite num ber of dimensions. The unknown functions Q , δ and φ are calculated through the solution of three nonlinear ordinary differential equations. In the follow- ing, we present only the final forms of the solutions, the metho d b eing thoroughly discussed in [45, 48, 49]. The first-order solutions are in general doubly p erio dic, i.e. p erio dic b oth in time and space and depend on three real parameters. W e classify the solutions in tw o types the same wa y as in [47] (see Fig. 1 of [47]). The type A solutions describe shifted recurrence, where local maxima in time app ear with a shift of half temp oral p erio d after a propagation distance corresp onding to a half of spatial p eriod. The type B solutions describ e the recurrence, where the local maxima in time appear at the same tem- p oral position after a propagation distance corresponding to one spatial p erio d. The separatrix b etw een these tw o families, kno wn as Akhmediev breather, is a hetero clinic orbit (p eriodic in time, non p erio dic is z ), connecting the t wo CW solutions of the same amplitude but different phases.The conv en tion taken in the recent work [50] is the same as here. How ev er, the notation used in [11] is rev ersed. The family of solutions is controlled by three parame- ters α 1 , α 2 and α 3 whic h are the three ro ots of a fourth order p olynomial, with the fourth one b eing zero [48]. They can either b e all real, ordered in suc h a w a y that α 3 > α 2 > α 1 > 0, or one real and tw o complex conju- gates: α 3 > 0 and α 1 = α ∗ 2 = ρ + iη . Division of the the ro ots α i b y some p ositive n um b er C is equiv alen t to transition from the solution ψ ( t, z ) corresp onding to the ro ots α i to a different solution ψ 0 ( t, z ) corresp onding to the ro ots α i /C . The tw o solution are connected by the transformation ψ ( t, z ) = C ψ 0 ( C t, C 2 z ) . (3) As a constan t C , w e can c ho ose, for example, the v alue of one of the ro ots and seek a tw o-parameter family of solu- tions. The third parameter can be rein troduced b y means of the transformation (3), thus adjusting the amplitude. In the following, w e consider α 3 = 1, whic h physically means fixing the initial CW component of the solution (strictly for the A-t yp e solutions and approximately for the B-type). The tw o parameters α 1 and α 2 allo w us to tune indep endently the spatial and temp oral p erio ds of the solutions. In practice, this means that if the p eriod and amplitude of the initial condition are known, we can predict the oscillating pattern of the resulting ev olution. A. The B-type solutions As explained ab ov e, here, we stick to the definition of doubly p erio dic solutions in tro duced in [47]. Namely , A-t yp e solutions are those lo cated inside the separatrix corresp onding to the Akhmediev breather while B-type solutions are lo cated outside of the separatrix. W e start with the B-t yp e solutions. They dep end on the three real parameters α 3 > α 2 > α 1 > 0. 3 F or the function δ ( z ), w e ha v e the follo wing expression: δ ( z ) = √ α 1 α 3 sn( µz , k ) p α 3 − α 1 cn 2 ( µz , k ) , (4) where the mo dulus m of the Jacobian elliptic func- tions [54, 55] is m = k 2 = α 1 ( α 3 − α 2 ) α 2 ( α 3 − α 1 ) and µ = 2 p α 2 ( α 3 − α 1 ). It is important to note that 0 < δ 2 < α 1 b y construction. F or the function φ ( z ), w e hav e the follo wing expression: φ ( z ) = ( α 1 + α 2 − α 3 ) z + 2 α 3 µ Π(am( µz , k ) , n, k ) , (5) where n = α 1 α 1 − α 3 and Π(am( µz , k ) , n, k ) is the incom- plete elliptic in tegral of the third kind, with the argumen t am( u, k ) b eing the amplitude function [54, 55]. F or the function of t wo v ariables Q ( t, z ), we hav e the follo wing expression: Q ( t, z ) = Q D ( Q A − Q C ) + Q A ( Q C − Q D )sn 2 ( p t, k q ) ( Q A − Q C ) + ( Q C − Q D )sn 2 ( p t, k q ) , (6) where the elliptic mo dulus is m q = k 2 q = α 2 − α 1 α 3 − α 1 and p = √ α 3 − α 1 . Importantly , Q C < Q < Q D b y con- struction. The z -dep enden t functions Q A ( z ) > Q B ( z ) > Q C ( z ) > Q D ( z ) are defined by the following expressions: Q A = s √ α 1 − y + √ α 2 − y + √ α 3 − y , (7) Q B = − s √ α 1 − y − √ α 2 − y + √ α 3 − y , (8) Q C = − s √ α 1 − y + √ α 2 − y − √ α 3 − y , (9) Q D = s √ α 1 − y − √ α 2 − y − √ α 3 − y , (10) where y ( z ) = δ 2 ( z ), and s = s ( z ) = sign( δ z ) = sign(cn( µz , k )). The expressions in Eqs. (7)-(10) differ from the original ones given in [48] by the sign function s in the first terms. This amendment remov es the dis- con tinuities of the z -deriv ativ e of the field ψ , which were presen t in the original formulation. The perio d along z (= L ) and along t (= T ) can b e calculated as follows: L = 4 K ( k ) µ , T = 2 K ( k q ) p , (11) where K ( k ) is the complete elliptic in tegral of the first kind [54]. A typical example of the B-t yp e solution is sho wn in Fig. 1. P arameters of the solution are given in the figure caption. Figure 1(a) shows the spatio-temporal ev olu- tion of the intensit y | ψ | 2 , and Fig. 1(c) shows the in- tensit y profile at the input ( z = 0, red curv e) and at the p oin t of the maximum temp oral pulse compression ( z = L/ 2 ≈ 3 . 8, blue curv e). With the given set of pa- rameters, the solution is similar to the ev olution of the separatrix (AB) up to z ≈ 7 . 5. It describ es the am- plification of a small but finite perio dic mo dulation on top of a strong CW to wards the generation of a p erio dic train of pulses. Just as the separatrix, after the max- im um compression, the field returns back to its initial profile. Ho w ever, in contrast to the separatrix, this hap- p ens at finite length z = L = 7 . 55. In sp ectral domain, the energy spreads from the central comp onen t to side- bands with the following return back to the central one. W e hav e, thus, an analytic description of the FPU recur- rence phenomenon except that the initial spectrum is not completely concentrated in a single comp onen t. An im- p ortan t p oin t is that the presence of three indep endent parameters of the family allows us to control indep en- den tly p erio ds in x and t and the amplitude of the input. This is discussed, in more detail, b elow. (a) (b) (c) (d) FIG. 1: An example of the B-type double-p eriodic solution. (a) F alse colour plot of intensit y | ψ ( z , t ) | 2 . Two longitudinal p eriods are shown. (b) Evolution of the sp ectrum. (c) Inten- sit y profile | ψ ( t ) | 2 at z = 0 (red curv e, minim um of modu- lation) and at z = L/ 2 = 3 . 78 (blue curve, maximal mo du- lation). (d) Evolution of the first three F ourier comp onen ts in logarithmic scale (20 log 10 | ˆ ψ k ( z ) | ). P arameters: α 1 = 0 . 3, α 2 = 0 . 4, α 3 = 1, giving the temp oral perio d T = 3 . 9 (mo d- ulation frequency ω = 2 π /T = 1 . 61) and the spatial p erio d L = 7 . 55. B. F ourier sp ectrum of the B-type solution F rom practical p oin t of view, imp ortant parameters of p erio dic solutions are their sp ectral comp onen ts. Re- mark ably , for the ab o ve solutions, the calculations can b e done analytically . Being p erio dic in t v ariable, the Q function in Eq. (6) can b e expanded in F ourier series as 4 follo ws: Q ( t, z ) = + ∞ X ` = −∞ ˆ Q ` ( z ) e i 2 π` T t = ˆ Q 0 ( z ) + 2 + ∞ X ` =1 ˆ Q ` ( z ) cos  2 π ` T t  , (12) where the z -dependent F ourier co efficien ts [56] are: ˆ Q 0 ( z ) = Q D + ( Q D − Q A ) Π( n, k q ) K ( k q ) , ˆ Q ` ( z ) = ( Q D − Q A ) π λ 2 K ( k q ) sinh(2 `w ) sinh(2 `w 0 ) . (13) (14) Here, n = ( Q D − Q C ) / ( Q A − Q C ) , λ = r n ( n − 1)( m q − n ) , w = π [ K ( k 0 q ) − v 0 ] 2 K ( k q ) , w 0 = π K ( k 0 q ) 2 K ( k q ) , v 0 = F (sin − 1 (1 − n ) − 1 / 2 , k 0 q ) , k 0 2 q = 1 − k 2 q and F ( ϕ, k ) is the incomplete elliptic in tegral of the first kind [54, 55]. The ev olution of the F ourier co efficien ts for the total field ψ ( x, t ) is simply given by: ˆ ψ 0 ( z ) = [ ˆ Q 0 ( z ) + iδ ( z )] e iφ ( z ) , (15) ˆ ψ ` ( z ) = ˆ Q ` ( z ) e iφ ( z ) . (16) The evolution of the sp ectrum of the B-type solution for the same set of parameters as in Fig. 1(a) is shown in Fig. 1(b ). The F ourier spectrum is symmetric with resp ect to ω = 0. F or the c hoice of parameters in Fig. 1, the initial p ow er of the CW comp onent is appro xi- mately 1 (0 dB). The total range of changes shown here is 80 dB. Fiv e F ourier components of the initial profile are lo cated within this range. Namely , | ˆ ψ 0 (0) | ≈ 0 dB, | ˆ ψ ± 1 (0) | = − 27 . 5 dB, | ˆ ψ ± 2 (0) | = − 57 . 2 dB. At the point of maximal pulse compression, the solution b ecomes a com b of 17 sp ectral lines that has a triangular shap e. In- deed, the p o w er of the harmonics deca ys as a geometrical progression with the order, as can b e seen from Eq.(14). As the solution is z -p erio dic, it describ es an energy cas- cade to w ards higher harmonics, whic h is rep eatedly re- v ersed back to the original sp ectrum. The detailed evo- lution of the first three sp ectral comp onents is sho wn in Fig. 2(c). F rom this figure and from Eq.(14), it can b e seen that the sidebands never v anish, i.e. the solution nev er turns to CW with | ψ | = constant. This only hap- p ens for the sp ecial c hoice of parameters when the doubly p eriodic solution approaches the separatrix. C. Ma jor c haracteristics of the B-t yp e solutions As mentioned, the family of the B-type solutions has 3 v ariable parameters. Only tw o of them can b e con- v eniently shown on a plane. Th us, we keep the scaling parameter α 3 fixed. It relates the amplitude and the tw o p eriods along the z and t axes. Th us, it can b e used in practical calculations for adjusting any of these pa- rameters to the actual v alues obtained in exp erimen ts. Here, w e present the t w o p erio ds (or the corresponding frequency and wa v en umber) and the maximal amplitude for fixed α 3 = 1. W e recall that parameters α 1 and α 2 v ary in the in- terv als of v alues from 0 to α 3 = 1. Moreo v er, we hav e assumed that the ro ots are ordered in such a wa y that α 2 > α 1 . Th us, it is sufficient to consider these pa- rameters within the triangular area shown in colour in Figs.2(a) and 2(b). Figure 2(a) sho ws the frequency ω = 2 π/T while Figure 2(b) shows the wa v enum ber q = 2 π/L b oth calculated using Eq.(11). The crucial observ ation is that the range of admitted frequencies for the B-t yp e solutions coincides with the mo dulational in- stabilit y band 0 < ω < 2. Then, it is not surprising that the amplification of small p erio dic perturbations as a ma jor feature of mo dulation instability remains v alid at small deviations from the separatrix as we hav e seen in the previous Subsection. AB DN KM PS CW BS (a) (b) (c) (d) DP FIG. 2: Ma jor characteristics of the B-type doubly p erio dic solutions on the plane ( α 1 , α 2 ) when α 3 = 1. (a) F alse color plot of frequency ω = 2 π /T . (b) F alse color plot of wa ven um- b er q = 2 π /L . The thin curves in (a) and (b) show the lines of equal frequency or wa ven um b er. (c) Maximal and minimal amplitudes of | ψ ( t = 0 , z ) | . (d) Absolute v alue of the CW comp onen t in ψ ( x, t ) at z = 0. Sp ecial cases in (a): PS - P ere- grine solution, CW - the family of con tinuous wa v e solutions, DN - the family of DNoidal wa ve solutions, AB - the family of Akhmediev breathers, KM - the family of Kuznetsov-Ma solitons, BS - brigh t soliton on zero background. DP - the t wo parameter family of doubly p erio dic solutions. Ho wev er, the family of the B-type solutions includes 5 m uch wider set of solutions. When α 1 = α 2 , i.e. on the diagonal line in Figs.2(a,b), the solution b ecomes homo- clinic or a separatrix, kno wn as Akhmediev breathers. The members of this family of solutions ha ve finite p eri- o ds in t and infinite perio ds in z . T o be sp ecific, p erio d L in this limit tends to infinity . A t the p oin t α 1 = α 2 = 1, the solution degenerates to the Peregrine soliton. In this case, b oth p eriods are infinite meaning that the solution is lo calized b oth in time and in space. The vertical line α 2 = 1 corresp onds to the Kuznetsov-Ma s oliton or a soliton on a finite background. This solution is p erio dic in z and lo calised in t . The background b ecomes zero at the p oint α 1 = 0 and α 2 = 1. Thus, the soliton on a bac kground turns into an ordinary bright soliton on zero bac kground. The line α 1 = 0 corresp onds to the p eri- o dic in t and stationary in z solutions known as DN oidal w av es. The p erio d is v ariable and changes along the hor- izon tal line. The p oin t α 1 = α 2 = 0 corresp onds to the CW solution. The straight line connecting the p oints (1/2,1/2) and (0,1) on the plane ( α 1 , α 2 ) corresp onds to the family of doubly p erio dic solutions first presen ted in [47] (see Eq.(18) of this work). It was further studied theoretically and experimentally in more recent works [11, 50]. F or p eriodic solutions, imp ortant physical parameters are the maximal and the minimal amplitudes of the w a v e profiles. W e present here the minim um and the maximum v alues of ψ ( z , 0) as a function of distance z at fixed time t = 0. The t = 0 is chosen as the p oint where the wa v e amplitude c hanges the most. This can be seen clearly from Fig.1(a). In particular, the ratio of these v alues can serv e as a measure of the contrast of amplitude oscilla- tions. F rom the exact solution, we get: ψ min = min z | ψ ( t = 0 , z ) | = − √ α 1 + √ α 2 + √ α 3 , (17) ψ max = max z | ψ ( t = 0 , z ) | = √ α 1 + √ α 2 + √ α 3 . (18) The tw o surfaces describ ed by Eqs.(17-18) are shown in Fig. 2(c). Naturally , they are joined together at the line α 1 = 0 as the D N oidal wa ve do es not evolv e in time. The minimal amplitude here is 1. The maximal amplitude of 3 is reached in the case of the P eregrine solution ( α 1 = α 2 = 1). This is the expected absolute maxim um of the AB solutions at this limit. F or completeness, Fig.2(d) shows the amplitude of the CW component at z = 0 extracted from Eq.(13). F or the AB solutions, the CW comp onent is alwa ys 1 when α 3 = 1. F or all other cases, the CW comp onent is smaller. More generally , when α 3 is differen t from 1, the estimates can b e made in terms of α 3 . F or v alues of parameters sufficien tly far from the cases α 1 = 0 (DN limit) and α 2 = 0 (KM limit), the estimate is | ˆ ψ 0 (0) | ≈ √ α 3 . This means that for the solutions describing the amplification of a small harmonic p erturbation ov er a strong CW, the v alue of the CW is fixed mainly by the parameter α 3 . D. The family of A-type solutions As sho wn in [48], A-t yp e solutions, α 1 and α 2 are com- plex and it is more conv enien t to switc h to other tw o pa- rameters ρ and η defined as α 1 = α ∗ 2 = ρ + iη . Then the family of solutions dep ends on three real parameters α 3 > 0 , ρ, η . In this case, for the function δ ( z ), w e hav e the following expression: δ ( z ) = r α 3 2 (1 − ν ) s 1 + dn( µz , k ) 1 + ν cn( µz , k ) sn( µz / 2 , k ) , (19) where m = k 2 = 1 2  1 − η 2 + ρ ( ρ − α 3 ) AB  , A 2 = ( α 3 − ρ ) 2 + η 2 , B 2 = ρ 2 + η 2 , ν = A − B A + B , and µ = 4 √ AB . Clearly , 0 < δ 2 < α 3 , by construction. The phase φ ( z ) is given b y: φ ( z ) =  2 ρ + α 3 ν  z − α 3 ν µ  Π(am( µz , k ) , n, k ) − − ν σ tan − 1  sd( µz , k ) σ   (20) where n = ν 2 ν 2 − 1 , σ = s 1 − ν 2 k 2 + (1 − k 2 ) ν 2 , and sd( µz , k ) = sn( µz , k ) dn( µz , k ) . F orm ula (20) for the phase is the corrected version of the one presented ea rlier in [45, 48]. The function Q ( t, z ) has the following expression: Q ( t, z ) = sb − c + r + cn( pt, k q ) 1 + r cn( pt, k q ) , (21) where s = s ( z ) = sign [cn( µz / 2 , k )], r = M − N M + N , p = √ M N = 2 4 p ( α 3 − ρ ) 2 + η 2 , k 2 q = 1 2 + 2 ρ − α 3 p 2 , b = √ α 3 − y , y ( z ) = δ 2 ( z ) , c ± = r 2 h p ( y − ρ ) 2 + η 2 ± ( ρ − y ) i , M 2 = (2 sb + c + ) 2 + c 2 − , N 2 = (2 sb − c + ) 2 + c 2 − . The p erio ds along z (= L ) and t (= T ) are: L = 8 K ( k ) µ , T = 4 K ( k q ) p . (22) 6 A typical example of the intensit y profile evolution of A- t yp e doubly p erio dic solution is shown in Fig. 3(a). F or giv en set of parameters, this solution also describ es the amplification of a weakly mo dulated CW ( z = − L/ 4 = − 3 . 4) to wards the generation of a perio dic train of pulses ( z = 0). This can b e seen by comparing the initial (at z = − L/ 4) and the maximally compressed (at z = 0) profiles in Fig.3(c). Ho wev er, in contrast to the B-type solution, after the first recurrence back to the initial condition, where the initial CW state is recov ered (at z = L/ 4 = 3 . 4), the follow up evolution differs qualitativ ely from the B-type case. Namely , the next gro wth-decay cycle generates a train of pulses which is shifted by a half of the temp oral p erio d in time domain ( z = L/ 2 = 6 . 8) relativ e to the pulse train in the first cycle. Snapshots of the initial ( z = − L/ 4) field profile and the generated train of pulses ( z = 0) are shown in Fig. 3(c) by the red and the blue curves, respectively . Our analysis provides an elegan t analytic description of the FPU recurrence and a symmetry breaking in infinite- dimensional dynamical systems. Here, it takes the form of the transition b et w een the A-type and B-type doubly p eriodic solutions. Indeed, the spreading of the spectrum from one CW comp onent to several sidebands and the follo w up compression of the energy bac k to the same single comp onen t is the manifestation of the FPU re- currence. Switching b et w een the t wo scenarios of this recurrence while crossing the separatrix is a symmetry breaking. How ev er, w e should remember that the tran- sition from one type of orbits to another one occurs in an infinite-dimensional phase space. This transition is far from b eing a simple copy of symmetry breaking in systems with one degree of freedom. Complexity of these transitions can be seen from some examples given in [50]. The freedom of tuning indep endently the spatial and temp oral p erio ds of the solution is a p o werful to ol in de- scription of v ariet y of physical phenomena. In addition, the third parameter α 3 pro vides the freedom of changing arbitrarily the amplitude of the solutions. Clearly , this family of solutions is more general than the one used in [11]. E. F ourier sp ectra of A-type solutions The F ourier coefficients of Q function giv en b y Eq. (21) can b e calculated using the technique from [56]. These sp ectra are as follo ws: ˆ Q 0 ( z ) = sb + c + r  Π( n, k q ) K ( k q ) − 1  , ˆ Q ` ( z ) = c + π λ 2 K ( k q ) ·        s sinh( `w ) sinh( `w 0 ) , if ` is even − cosh( `w ) cosh( `w 0 ) , if ` is o dd (23) (24) (a) (b) (c) (d) FIG. 3: An example of the A-type doubly p erio dic solution. (a) F alse color plot of intensit y | ψ ( t, z ) | 2 . One complete p e- rio d of evolution is shown. (b) Ev olution of the sp ectrum. (c) The in tensity profile, | ψ ( t ) | 2 , at z = − L/ 4 (red curve, the minimal mo dulation) and at z = 0 (blue curve, the maximal pulse compression). (d) Evolution of the first three F ourier comp onen ts in logarithmic scale (20 log 10 | ˆ ψ k ( z ) | ). P arame- ters: α 3 = 0 . 3, ρ = 0 . 355, η = 0 . 073. The resulting temp o- ral p erio d T = 3 . 9 corresp onds to the mo dulation frequency ω = 2 π /T = 1 . 61. The spatial p eriod L = 13 . 6. where n = r 2 / ( r 2 − 1), λ = s 1 − r 2 m q + r 2 (1 − m q ) , w = π [ K ( k 0 q ) − v 0 ] 2 K ( k q ) , w 0 = π K ( k 0 q ) 2 K ( k q ) , and v 0 = F (sin − 1 √ 1 − r 2 , k 0 q ). In terestingly , the even and o dd F ourier co efficients in (24) are different. This is in striking con trast to the case of the B-type solutions. An example of the ev olution of the sp ectra of A-type solution is shown in Fig. 3(b). The evolution of the p ow er sp ectrum is p erio dic in z with p erio d L/ 2. It sho ws p eriodic expansion to wards higher order spectral comp onents and recurrence back to the initial spectrum. The sp ectra remain symmetric with resp ect to zero frequency at every z . At the p oint of maximal expansion, a triangular comb of 17 lines can b e seen within the range of 80 dB. These include the cen tral comp onent | ˆ ψ 0 (0) | = 0 dB, and the first sideband | ˆ ψ ± 1(0) | = − 27 . 0 dB. The pair of second order sidebands v anishes, | ˆ ψ ± 2 (0) | = 0 = −∞ dB. Moreov er, all even or- der sidebands also completely v anish as can b e seen from Eq.(24). They are recov ered at the p oin ts of minimal sp ectral expansion ( z = L/ 4 + `L/ 2, ` = 0 , ± 1 , . . . ). This can b e seen in Fig. 3(d) whic h shows the ev olution of the first three sp ectral comp onen ts. 7 F. Ma jor c haracteristics of A-type solutions The ma jor characteristics of the family of doubly p eri- o dic A-type solutions are p eriods along the z and t axes and the wa v e amplitudes. In this section, w e will express them as a function of control parameters of the family ρ, η , α 3 . As b efore, w e keep α 3 = 1 fixed. The ampli- tude can b e rescaled when needed using Eq.(3). When α 3 = 1, the av erage initial amplitude | ˆ ψ 0 (0) | = 1, as it can b e seen from Eq.(23). Physically , this means that at the p oint of minimal spectral expansion, the A-type solution describ es a weak mo dulation of a CW of unit amplitude. Figures 4(a) and 4(b) sho w the frequency ω = 2 π /T and the wa v en umber q = 2 π /L , resp ectively , calculated from Eq.(22). The crucial difference from the B-t ype case is that there is no frequency cut-off for the A-type solu- tions. This means that these solutions can describ e the gro wth of weak p erturbations on a constant bac kground ev en outside the c onventional MI b and , [0,2]. The whole y ellow-red area in Fig.4(a) corresp onds to the frequen- cies ab o ve the limiting v alue 2. Comparing Figs.4(a) and 4(b), we conclude that, in this region, the higher the fre- quency ω , the higher is the wa v en umber q . Thus, when frequency ω is well outside the MI band, the solution will oscillate rapidly in z direction. This is a typical feature of non phase-matched four-wa v e mixing. KM x CW AB PS (a) (b) (c) (d) FIG. 4: Ma jor c haracteristics of A-type doubly perio dic solu- tions vs ρ and η . F alse color plots of (a) frequency ω = 2 π /T and (b) w av en umber q = 2 π /L . (c) Maximal ( ψ max ) and minimal ( ψ min ) v alues of | ψ ( t = 0 , z ) | . (d) Difference betw een the maximal and minimal v alues of | ψ ( t = 0 , z ) | . Notations in (a): PS - Peregrine solution, CW - con tinuous wa v e, AB - Akhmediev breathers, KM - Kuznetsov-Ma solitons, BS - brigh t soliton. Here parameter α 3 = 1. P articular cases of this general three-parameter family of solutions can b e identified based on spatial and tem- p oral p erio ds. Each of them is still a family of solutions with lo w er n um b er (tw o) of parameters. When η = 0 and 0 < ρ < 1, w e get the homo clinic AB solutions. They are p eriodic in t while p eriod L along z -axis tends to in- finit y . When η = 0 and ρ = 1, the solution b ecomes the P eregrine soliton with no free parameters (except α 3 ). It is lo calized b oth in time and in space thus representing a rogue w a ve. The case η → 0 and ρ > 1 corresp onds to the Kuznetsov-Ma solitons. They are p erio dic in z and lo calized in t . When η = 0 and ρ < 0, the solution b ecomes the trivial CW (homogeneous) solution. Note that if η is strictly zero, η = 0, then ρ < 1 b ecause the ro ots α 1 = α 2 = ρ < α 3 b y definition. The minimum and the maximum of ψ (0 , z ) as a func- tion of distance for t = 0 are given b y: ψ min =      √ α 3 − r 2 h p ρ 2 + η 2 + ρ i      , (25) ψ max = √ α 3 + r 2 h p ρ 2 + η 2 + ρ i . (26) The tw o surfaces described by Eqs.(25) and (26) are sho wn in Fig.4(c). It can be seen from Fig.4(c) that when p ρ 2 + η 2 + ρ > α 3 / 2, the tw o surfaces are nearly parallel. Then the difference b etw een the maximal and minimal v alues saturates to the maximum v alue ψ max − ψ max = 2 √ α 3 . This is shown separately in Fig.4(d). I II. NONLINEAR ST A GE OF LINEARL Y ST ABLE PER TURBA TIONS The fact that the A-type solutions ha v e an arbitrary temp oral p erio d, ha v e in triguing effects on the nonlinear stage of MI. As we kno w w ell [5, 6], perio dic perturba- tions outside the con v entional MI band, 0 < ω < 2, in the linearised problem are not growing. The detailed analy- sis of A-type solutions leads to a different conclusion, as explained b elo w in detail. Figure 5 shows the amplification of the first F ourier harmonic for B-type and A-type solutions. Namely , Fig. 5(a) is a 2D colour coded plot of gain of the first sideband defined as G = | ˆ ψ 1 ( L/ 2) / ˆ ψ 1 (0) | in dB for the B-t yp e so- lution as a function of parameters ( α 1 , α 2 ) when α 3 = 1. On the line α 1 = α 2 , the gain defined this wa y , go es to infinit y , b ecause the mo dulation tends to zero for z → 0 and the p erio d L → ∞ . In order to enhance the read- abilit y of the picture, the colour scale is saturated at 40 dB. The lines of constan t frequency ω are sup erimp osed in order to see their v alue relative to the MI band. The gain is decreasing aw ay from the AB limit, and com- pletely v anishes on the line α 1 = 0. The solution at this line describ es a z -stationary solution. Three gain curves along the lines α 1 = α 2 − ∆ a parallel to the line of AB solutions are shown in Fig. 5(c). The highest amplifi- cation is observed on the line that is closest to the AB limit. As these curves sho w, all frequencies within the MI band are amplified as w e would expect from the stan- dard MI analysis. Gain is zero outside of this band also in agreement with the standard MI theory . 8 (a) (b) (c) (d) FIG. 5: F alse colour plots of amplification of the first F ourier harmonics | ˆ ψ 1 | in decib els for (a) the B-type and (b) the A-t yp e doubly p eriodic solutions. The superimp osed level curv es are for the constant frequencies ω . Colormap is sat- urated at 40 dB for readability . (c) Amplification curves on selected straight lines α 1 = α 2 − ∆ a in the plot (a). Here ∆ a < α 2 < 1. (d) Amplification curves at selected lines of constan t η in the plot (b). Here − 3 < ρ < 1. V ertical dashed lines in (c) and (d) show the MI threshold for a CW of unit amplitude. In all cases, α 3 = 1. The situation is differen t for the A-t yp e solutions. Fig- ure 5(c) sho ws the colour co ded 2D plot of gain defined the same w a y G = | ˆ ψ 1 (0) / ˆ ψ 1 ( − L/ 4) | in dB for A-type solution as a function of parameters ( ρ, η ). The third pa- rameter α 3 = 1. The lines of constan t frequency ω are sup erimposed on this plot allo wing us to iden tify the fre- quencies inside and outside the MI band. The red curve in this plot corresp onds to the upp er limit of conv en- tional MI band, ω = 2. As in the case of the B-t yp e solutions, the gain tends to infinity within the conv en- tional MI band (0 < ω < 2) when η → 0 and 0 < ρ < 1. The z -p erio d on this line is also infinite as it should b e for the AB solutions. The colour scale here is the same as in Fig. 5(a). It is saturated at 40 dB. The v alue of gain decreases aw a y from this area, as exp ected. How ev er, it never v anishes to zero which is an imp ortan t conclu- sion of our analysis. Indeed, this is the most striking feature of A-type solutions. They describ e amplific ation and p ar ametric gain outside the standar d MI b and . This feature is further evident from Fig. 5(d), showing gain curv es at small constant v alues of η . As we can see from these c urv es, there is no cut-off at ω = 2, but rather a smo oth transition to a region of small gain for ω > 2. This gain slowly drops at large frequencies, b eing nearly indep enden t on η . An example of the A-t yp e solution illustrating the phe- nomenon of the first sideband amplification outside the instabilit y band and the follow up recurrence is shown in Fig. 6. The evolution starts with CW field with small but finite nearly sinusoidal p erturbation. This initial condi- (a) (b) (c) (d) FIG. 6: Evolution of the A-type doubly p eriodic solution il- lustrating the amplification outside the conv en tional MI band. (a) F alse color plot of in tensity | ψ ( z , t ) | 2 . (b) Evolution of the first three F ourier components. (c) Intensit y profiles | ψ ( t ) | 2 at z = − L/ 4 (red curve, minimum of mo dulation) and at z = 0 (blue curve, maxim um pulse compression). (d) Input (red) and output (blue) spectra. The input sp ectrum has a pair of small second sidebands. Parameters of the solution: ρ = 0, η = 1, α 3 = 1. In this case, the temp oral p erio d T = 2 . 74 (mo dulation frequency ω = 2 . 287 > 2) and the spatial p erio d L = 2 . 75. tion is sho wn b y red curv e in Fig. 6(c). The amplitude of mo dulation grows transforming the wa v e profile to a train of pulses. The maximum amplitude of pulses is reac hed after a quarter of the z -p erio d at the p oint z = 0. The transformed profile is shown as blue curve in Fig. 6(c). P erio dicit y tak es the profile bac k to the initial shap e after a half of the z -p erio d. This occurs at z = L/ 4. How ev er, the phase of the sinusoid is no w shifted in time. The next pulse compression p oint is z = L/ 2. The maxima of the pulses are now lo cated at the p osition of minima of the previously compressed profile although the shap e remains the same. The e v olution of the low est order F ourier comp onents during this pro cess is shown in Fig. 6(b). Amplification of the first sideband in the first quarter of the p erio d is clearly seen. Its pow er at the p oint of maxim um increases b y g = | ˆ ψ 1 (0) / ˆ ψ 1 ( − L/ 4) | 2 = 2 . 19 = 3 . 4 dB. This hap- p ens at the exp ense of significantly depleted CW com- p onen t. The pro cess is p erio dic thus leading to p erio dic ev olution of sp ectral comp onents. Figure 6(d) shows the sp ectral conten t of the initial condition (red circles) and the maximally compressed pulse train profile (blue cir- cles). A remark able observ ation that may lead to im- p ortan t applications is that ev en harmonics are zero at the input field. This is clearly seen in Fig.6(d). They are generated in ev olution and become the strongest at the p oint of maximal com pression still keeping the total sp ectrum within the triangular shape (blue points). In the time domain, this sp ectral feature corresp onds to the 9 p eriod-doubling of the field intensit y . This can b e seen clearly in Fig. 6(c). Period of the blue curve is twice the p eriod of the red curve. This is different from the case sho wn in Fig.1(c) where p erio ds of the initial condition and the resulting train of pulses coincide. Clearly , the fre- quency is halv ed when perio d doubling takes place. This effect is opposite to the frequency doubling phenomenon considered earlier in [57] once again demonstrating the ric hness of phenomena contained in the family of doubly p eriodic solutions. The example shown in Fig. 6(b) corresponds to a case where the initial amplitude of the first-order sideband is relatively large. One migh t wonder, ho w small the sideband could b e to still achiev e amplification out-of- instabilit y-band. In order to answer this question and further rev eal the difference b etw een the b ehaviors of A- t yp e and B-type solutions concerning the out-band am- plification, we sho w, in Fig.7, the minimum v alue ov er z of the first-order F ourier harmonic | ˆ ψ 1 | . This corresponds indeed to the smallest sideband that can b e amplified (along with the harmonics that are inv olv ed in the solu- tions) for an y giv en choice of the parameters α 1 , α 2 for B-t yp e, or ρ, η for A-type solutions, assuming α 3 = 1 as b efore. Figure 7(a), whic h is related to t yp e B solutions, sho ws that the sideband amplitude is defined only inside the conv en tional MI band ω < 2, ranging from arbitrar- ily small v alues (close to AB, α 1 ' α 2 ) to relatively large v alues when the solution is a deformation of DN-oidal solution ( α 1 = 0, see Fig. 2) with large α 2 . As shown in Fig. 7(b), for the A-type solutions, small sideband am- plitudes are obtained close to the horizontal axis ( η ' 0) for ρ < 1 where, at v ariance with previous case, they ex- ist across the band edge limit (parametric curve ω = 2) near the origin of the plane ( ρ, η ). As a consequence, one can still amplify even small sidebands, although it must b e considered that the gain abruptly drops across suc h threshold (see Fig. 5(b) and the blue curve in Fig. 5(d)), whereas the p eriod L decreases. This regime is similar to a parametric amplification c haracterized by large phase mismatc h. Larger v alues of η correspond, in the out- band region, to a rapid increase of the sideband to v alues whic h are typically larger than those for B-type solution, and can b ecome comparable to the amplitude of the CW. This is the regime of amplification of large mo dulations, whic h, b y definition, cannot be described b y the standard MI linear stability analysis. Finally , w e give an imp ortan t argument that helps to get insigh t into the fact that only A-t ype solutions ex- hibit gain outside the con ven tional MI gain bandwidth. T o this end, we resort to the in terpretation of MI as a symmetry breaking pro cess, where the broken symmetry app ears as the parameter ω is decreased b elow the thresh- old ω = 2 [12]. As also discussed ab ov e, the phase with brok en symmetry ( ω < 2) is characterized by the co exis- tence of A-type and B-type evolutions [12]. Conv ersely , for ω > 2, only one type of evolutions exist, which can b e considered, for ω  2, a small deformation of strictly line ar solutions. The latter can b e easily obtained from (a) (b) FIG. 7: F alse colour plots of minim um amplitude of the first F ourier harmonics | ˆ ψ 1 | in the plane of free parameters of the family of solutions: (a) B-type solutions (minimum sidebands obtained at z = 0; parameters α 1 , α 2 ); (b) A-t yp e solutions (minim um sidebands obtained at z = − L/ 4; parameters ρ, η ). Here α 3 = 1. the NLSE (1) by dropping the nonlinear term | ψ | 2 ψ . By considering, for the sake of simplicity , the initial condi- tion ψ ( t, 0) = ψ 0 + c 1 exp( iω t ) + c 1 exp( − iω t ) con taining the pump ψ 0 and a single symmetric pair of sidebands with complex amplitude c 1 , the linear solution reads as ψ ( z , t ) = ψ 0 +[ c 1 exp( iω t ) + c 1 exp( − iω t )] exp  − i ω 2 2 z  . (27) Equation (27) shows that the disp ersion is resp onsible for a contin uous phase rotation of the mo dulation, which go es through alternating states of amplitude and fre- quency modulation, characterised by relative phase be- t ween the sideband and the pump of 0 , π and ± π / 2, resp ectiv ely [58]. In particular, tw o successive states of amplitude mo dulation, obtained at relativ e distance ∆ z = 2 π /ω 2 (half spatial p erio d of solution in Eq. (27)) suc h that they are phase shifted by ω 2 ∆ z / 2 = π , exhibit a temp oral shift of half p eriod in the in tensit y pattern, similar to the case of A-type solutions. F rom this p oint of view, nonlinear solutions of A-type can b e considered as the nonlinear dressing of linear solutions, whic h exist for ω > 2 and are smo othly con tinued in to the phase with brok en symmetry ( ω < 2). Conv ersely , the B-type solu- tions hav e genuine nonlinear origin, bearing no analogy to linear solutions. Instead, they app ear only in conjunc- tion with the onset of MI, due to the symmetry breaking nature of the phenomenon. IV. CONCLUSION W e studied, in detail, the three parameter family of doubly p eriodic solutions of the nonlinear Schr¨ odinger equation, originally deriv ed b y Akhmediev, Eleonskii and Kulagin [48]. W e rev eal the richness of physical phenom- ena that is contained within this family . The three free parameters of this family allo w us to control arbitrarily the spatial and temp oral p eriods of the family and the amplitude of the resulting p erio dic profiles. 10 W e discov ered several new physical phenomena within this family . These include modulation instabilit y outside of the standard instabilit y band kno wn from the classi- cal works of Benjamin-F eir [5] and Bespalov-T alano v [6]. Using this expanded knowledge of modulation instability applied to the classical NLSE will lead to b etter under- standing of nonlinear phenomena and their new applica- tions b oth in optics and water wa v es. One of the ma jor adv ances of our presen t work is cal- culation of F ourier comp onen ts of p eriodic field in ex- plicit form. These analytic expressions will lead to fur- ther progress in physical applications of the NLSE related phenomena in fibre optics and water w a ves. F rom theoretical p oint of view, our analysis provides more physical understanding of fields generated by p e- rio dic initial conditions. Previous approaches based on general solution expressed in terms of theta functions did not lead to a significan t progress as it is difficult to extract ph ysically relev ant results from general solutions. It is a b etter idea to construct general p erio dic solutions from fundamen tal ones in the w a y similar to construction of m ulti-soliton solutions from its fundamental constituen ts - single solitons. There are sev eral techniques that can be used for this aim such as Darb oux or Bac klund transfor- mations. Being equipp ed with the family of the funda- men tal doubly p eriodic solutions of the NLSE, it will b e p ossible to construct more complicated solutions using this one as a starting p oin t. 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