The finite gap method and the periodic NLS Cauchy problem of the anomalous waves, for a finite number of unstable modes
TThe finite gap method and the periodic NLS Cauchy problemof the anomalous waves, for a finite number of unstable modes
P. G. Grinevich , and P. M. Santini , L.D. Landau Institute for Theoretical Physics, pr. Akademika Semenova1a, Chernogolovka, 142432, Russia, Dipartimento di Fisica, Universit`a di Roma ”La Sapienza”, andIstituto Nazionale di Fisica Nucleare (INFN), Sezione di Roma,Piazz.le Aldo Moro 2, I-00185 Roma, Italy e-mail: [email protected] e-mail: [email protected] November 22, 2018
Abstract
The focusing Nonlinear Schr¨odinger (NLS) equation is the simplestuniversal model describing the modulation instability (MI) of quasimonochromatic waves in weakly nonlinear media, and MI is consid-ered the main physical mechanism for the appearence of anomalous(rogue) waves (AWs) in nature. In this paper we study, using the fi-nite gap method, the NLS Cauchy problem for generic periodic initialperturbations of the unstable background solution of NLS (what wecall the Cauchy problem of the AWs), in the case of a finite number N of unstable modes. We show how the finite gap method adapts to thisspecific Cauchy problem through three basic simplifications, allowingone to construct the solution, at the leading and relevant order, interms of elementary functions of the initial data. More precisely, weshow that, at the leading order, i) the initial data generate a partitionof the time axis into a sequence of finite intervals; ii) in each interval I of the partition, only a subset of N ( I ) ≤ N unstable modes are“visible”, and iii) the NLS solution is approximated, for t ∈ I , bythe N ( I )-soliton solution of Akhmediev type, describing the nonlin-ear interaction of these visible unstable modes, whose parameters areexpressed in terms of the initial data through elementary functions.This result explains the relevance of the m -soliton solutions of Akhme-diev type, with m ≤ N , in the generic periodic Cauchy problem of theAWs, in the case of a finite number N of unstable modes. a r X i v : . [ m a t h - ph ] N ov Introduction
The two Nonlinear Schr¨odinger (NLS) equations iu t + u xx + 2 ν | u | u = 0 , u = u ( x, t ) ∈ C , ν = ± u ( x, t )is the complex amplitude of the electric field and the self-interacting term2 ν | u ( x, t ) | accounts for the nonlinear response of the medium, in a Kerrlike regime, proportional to the light intensity; in a quantum mechanicalinterpretation, u ( x, t ) is the wave function and V ( x, t ) = − ν | u ( x, t ) | isthe self-induced potential proportional to the probability density. If ν = 1,the self-interacting term is self-focusing (from which the name “self-focusingNLS”), and corresponds to a potential well whose depth increases with thedensity. If ν = −
1, the self-interacting term is defocusing (from which thename “defocusing NLS”), corresponding to a potential barrier whose heightincreases with the density. It is clear, from these considerations, that thefocusing and defocusing regimes correspond to very different evolutions ofthe same initial data.In particular, it is well-known that the elementary solution a exp(2 iν | a | t ) , a ∈ C and constant, (2)of (1), describing Stokes waves [96] in a water wave context, a state of con-stant light intensity in nonlinear optics, and a state of constant boson densityin a Bose-Einstein condensate, is stable, in the defocusing case, under the per-turbation of waves with arbitrary wave length, while it is unstable, in the fo-cusing case, under the perturbation of waves of sufficiently large wave length[23, 20, 107, 113, 99, 90]; this modulation instability (MI), present only in thefocusing case, is considered as the main cause for the formation of anomalous(rogue, extreme, freak) waves (AWs) in nature [54, 40, 84, 64, 65, 83]. In thispaper the term AW is deliberately used in a vague sense, and by it we justmean order one (or higher order) coherent structures over the unstable back-ground, generated by MI. This paper and our previous works [48, 49, 50, 51]2re dedicated to a complete analytic understanding of the deterministic as-pects of the dynamics of these coherent structures, in the case of a finitenumber of unstable modes.Let us point out that, in oceanography, anomalous (or rogue) waves aredefined as waves with a sufficiently high amplitude with respect to the av-erage one, and, in this respect, it is known that NLS soliton solutions overthe unstable background can reach unusually large amplitudes if their pa-rameters are specially correlated (see, for instance, [30]). The applicationof an amplitude criterion to select such anomalously large amplitude wavesamong finite-gap solutions of NLS over the background, was recently donein [21]. Our long-term research plan, actually one of the main motivationsof the studies made in this paper and in [48, 49, 50, 51], is to understandthe deterministic aspects of the theory, valid for a finite number of unstablemodes, as the starting point for the study of the statistical aspects of thetheory, when a large number of unstable modes are excited, in order to beable, in particular, to describe analytically the probability of the generationin space-time, due to MI, of anomalously large amplitude waves.The integrable nature [114] of the focusing NLS equation allows one toconstruct a large zoo of exact solutions, corresponding to perturbations of thebackground, by degenerating finite-gap solutions [62, 19, 69, 70], when thespectral curve becomes rational, or using classical Darboux transformations[77], dressing techniques [115, 111, 112], and the Hirota method [55, 56].Among these basic solutions, we mention the Peregrine soliton [85], ratio-nally localized in x and t over the background (2), the so-called Kuznetsov[71] - Kawata - Inoue [58] - Ma [75] soliton, exponentially localized in spaceover the background and periodic in time, and the solution found by Akhme-diev, Eleonskii and Kulagin in [11], periodic in x and exponentially localizedin time over the background (4), known in the literature as the Akhmedievbreather. Elliptic generalizations were constructed in [13]. A more generalone soliton solution over the background (2) can be found, f.i., in [62, 108],corresponding to a spectral parameter in general position. These solutionshave also been generalized to the case of multi-soliton solutions, describingtheir nonlinear interaction, see, f.i., [62, 37, 15, 109]. Finite-genus represen-tations of AWs were constructed in [21]; elliptic solutions corresponding tospecial genus 2 curves with symmetries were studied in [93] using the methodof [17]. We remark that the Peregrine solitons are homoclinic, describing AWsappearing apparently from nowhere and desappearing in the future, while themultisoliton solutions of Akhmediev type are almost homoclinic, returning to3he original background up to a multiplicative phase factor. Generalizationsof these solutions to the case of integrable multicomponent NLS equationshave also been found [18, 34, 35].Concerning the NLS Cauchy problems in which the initial condition con-sists of a perturbation of the exact background (2), what we call the Cauchyproblem of the AWs, if such a perturbation is localized, then slowly modu-lated periodic oscillations described by the elliptic solution of (1), for ν = 1,play a relevant role in the longtime regime [24, 25]. If the initial pertur-bation is x -periodic, numerical experiments and qualitative considerationsindicate that the solutions of NLS exhibit instead time recurrence [105, 73,106, 14, 103, 72], as well as numerically induced chaos [2, 8, 1], in whichthe almost homoclinic solutions of Akhmediev type seem to play a relevantrole [42, 45, 28, 29, 30]. There are reports of experiments in which the Pere-grine and the Akhmediev solitons were observed [31, 66, 106, 102, 59, 80, 87].Their relevance within some classes of localized initial data for NLS, in thesmall dispersion regime, was shown in [22, 41]; see also [47, 100] for theinvestigation of their relevance in ocean waves and fiber optics.The proper tool to solve the periodic Cauchy problem for soliton equationsis the finite gap method. Its development started in the papers [82, 38, 60,74, 78]; it was first applied to the NLS equation in [61], and its generalizationto 2+1 equations was first constructed in [68]. In the paper [48] and in thepresent work we apply it to solve the periodic Cauchy problem of the AWsfor the focusing NLS equation iu t + u xx + 2 | u | u = 0 , u = u ( x, t ) ∈ C ; (3)i.e., we study the focusing NLS Cauchy problem on the segment [0 , L ], withperiodic boundary conditions, for a generic, smooth, periodic, zero average,small initial perturbation of the background solution u ( x, t ) = e it ; (4)i.e.: u ( x,
0) = 1 + (cid:15)v ( x ) , < (cid:15) (cid:28) ,v ( x + L ) = v ( x ) , (5)where v ( x ) = ∞ (cid:88) j ≥ (cid:0) c j e ik j x + c − j e − ik j x (cid:1) , k j = 2 πL j. (6)We also assume that the period L be generic, i.e., that L/π is not an integer.4 emark 1
The simplified form (4) of the background (2) is obtained choos-ing a = 1 , without loss of generality, due to the scaling symmetry of NLS. In(6) we have also assumed, without loss of generality, due again to the scalingsymmetry of NLS, that the perturbation v ( x ) has zero average L (cid:90) v ( x ) dx = 0 , (7) implying that the Fourier coefficient c = (1 /L ) L (cid:82) v ( x ) dx be zero. It is well-known that a monochromatic perturbation of (4) of wave number k is unstable if | k | <
2; therefore, defining N ∈ N as N = (cid:22) Lπ (cid:23) , (8)where (cid:98) x (cid:99) , x ∈ R denotes the largest integer not greater than x , the first N modes {± k j } , ≤ j ≤ N , are linearly unstable, since they give rise toexponentially growing and decaying waves of amplitudes O ( (cid:15)e ± σ j t ), wherethe growing rates σ j are defined by σ j = k j (cid:113) − k j , ≤ j ≤ N, (9)while the remaining modes are linearly stable, since they give rise to smalloscillations of amplitude O ( (cid:15)e ± iω j t ), where ω j = k j (cid:113) k j − , j > N. (10)It is also convenient to introduce, for the unstable modes, the angles φ j ’sparametrizing them, and defined by φ j = arccos( k j /
2) = arccos (cid:16) πL j (cid:17) , < φ j < π/ , ≤ j ≤ N, (11)implying that k j = 2 cos φ j , σ j = 2 sin(2 φ j ) , ≤ j ≤ N. (12)5hese well-known facts are summarized in the following formula [48, 49],valid for 0 ≤ t ≤ O (1): u ( x, t ) = e it (cid:104) N (cid:80) j =1 (cid:16) (cid:15) | α j | sin 2 φ j e σ j t + iφ j cos[ k j ( x − x j )]+ (cid:15) | β j | sin 2 φ j e − σ j t − iφ j cos[ k j ( x − ˜ x j )] + O ( (cid:15) )-oscillations (cid:17)(cid:105) + O ( (cid:15) ) , (13)where α j = e − iφ j c j − e iφ j c − j , β j = e iφ j c − j − e − iφ j c j ,x j = arg( α j )+ π/ k j , ˜ x j = − arg( β j )+ π/ k j , j = 1 , . . . , N, (14)describing the first linear stage of MI, governed by the focusing NLS equationlinearized about the solution (4): δu t + δu xx + 2 exp(4 it ) δu + 4 δu = 0.Therefore the initial datum splits into exponentially growing and decaying waves, respec-tively the α - and β -waves, each one carrying half of the information encodedinto the unstable part of the initial datum, plus small oscillations associatedwith the stable modes, remaining small during the evolution .The j th unstable mode becomes O (1) at times of O ( σ − j | log (cid:15) | ); thereforethe most unstable modes, the ones appearing first, are the modes with largergrowth rate σ j . It follows that, at logarithmically large times, one entersthe (first) nonlinear stage of MI, the linearized NLS theory cannot be usedanymore, and, to describe the evolution, the full integrability machinery ofthe finite gap method for NLS must be used.Once applied to the specific Cauchy problem (3)-(7) of the AWs, we found,remarkably, that the finite gap method undergoes the following three basicsimplifications, allowing one to construct the solution, at the leading andrelevant order, in terms of elementary functions of the initial data (see [48]and this paper). Step 1: Finite-gap approximation.
A generic periodic O ( (cid:15) ) initialperturbation of the unstable background opens infinitely many O ( (cid:15) ) gaps inthe spectral problem. They are organized in pairs, and each pair correspondsto the pair ± k j of excited modes of the linearized problem. A finite number N = (cid:98) L/π (cid:99) of these modes are unstable, and the remaining ones are stable.Since the stable modes, giving rise to O ( (cid:15) ) oscillations, correspond to O ( (cid:15) )corrections to the AWs, one can close the corresponding gaps, keeping openonly the N gaps corresponding to the unstable modes (no matter how smallare these gaps, they will cause O (1) effects on the dynamics, due to the in-stability). Therefore, using this recipe, we go from an infinite-gap theory to6ts 2 N -gap approximation. Let us point out that this finite gap approxima-tion, specific to the Cauchy problem of AWs, is non standard. Indeed, inthe usual finite-gap approximation, one closes gaps smaller than a certainconstant, while, in our case, all gaps are small, and the criterion for closinga gap is the stability of the corresponding mode. Step 2. Explicit analytic approximation of the θ -function pa-rameters. The above finite-gap approximation of the problem implies thatthe solutions are represented by ratios of θ -functions of genus 2 N . In general,the parameters in the θ -function formulas are complicated transcendental ex-pressions in terms of the Cauchy data. In the special setting we use, goodelementary formulas in terms of the initial data can be written, at the lead-ing and relevant order, for all the parameters used in the arguments of the θ -functions . Step 3. Elementary approximation of the θ -functions. The Rie-mann θ -functions are defined as infinite sums of exponentials over all integerpoints in R N . Due to the presence of the small parameter (cid:15) , at each time itis sufficient to keep summation only over a subset of the 4 N vertices of theelementary hypercube of this multidimensional lattice containing the trajec-tory point. Therefore, for a generic t ≥ , the infinite sum of exponentialsreduces to a finite sum of N ( t ) exponentials, with ≤ N ( t ) ≤ N , whosearguments are given in terms of elementary functions of the Cauchy data.It turns out that this t -dependent representation of the solution in terms ofelementary functions coincides, at the leading order, with the N ( t ) -solitonsolution of Akhmediev type . Remark 2
We remark that the first attempt to apply the finite gap methodto solve the NLS Cauchy problem on the segment, for periodic perturbationsof the background, was made in [101]; the fact that, in the θ -function rep-resentation of the solution, different finite sets of lattice point are relevantin different time intervals was first observed there, but no connection wasestablished between the initial data and the parameters of the θ -function, andno description of the dynamics in terms of elementary functions was given. Using the above three simplifying steps, in [48] we have studied theCauchy problem of the AWs (3)-(6) in the particular case in which the initialperturbation excites just one of the unstable modes, say k n , ≤ n ≤ N : u ( x,
0) = 1 + (cid:15) ( c n e ik n x + c − n e − ik n x ) , ≤ n ≤ N, (15)7istinguishing two cases: 1) the case in which only the corresponding unstablegaps are opened by the initial condition (15), and 2) the case in which morethan one pair of unstable gaps is opened.In the first case 1), the solution describes an exact deterministic alternaterecurrence of linear and nonlinear stages of MI, and the nonlinear AW stagesare described by the Akhmediev breather, whose parameters, different ateach AW appearence, are always given in terms of the initial data throughelementary functions [48]. This result is summarized in the following Theorem 1
Consider the case in which the initial condition (15) opens onlythe pair of gaps associated with the corresponding modes ± k n of the linearizedproblem, and the associated finite gap Riemann surface is a genus N = 2 hyperelliptic curve with two O ( (cid:15) ) handles. This happens, for instance, if N = n = 1 ( π < L < π ), the simplest case in which only the mode k = 2 π/L isunstable, or if L/ π < n < L/π .Introduce the following parameters, for m ∈ N + : X ( m ) n = X (1) n + ( m − X n , T ( m ) n = T (1) n + ( m − T n , Φ ( m ) n = 2 φ n + ( m − φ n ,X (1) n = x n = arg( α n )+ π/ k n , ∆ X n = arg( α n β n ) k n , mod L,T (1) n = σ n log (cid:16) σ n (cid:15) | α n | (cid:17) , ∆ T n = σ n log (cid:18) σ n (cid:15) √ | α n β n | (cid:19) , (16) where α n and β n are defined in (14) in terms of the initial data.Then, for ≤ t ≤ O (1) , we are in the first linear stage of MI: u ( x, t ) = e it (cid:110) φ n ) (cid:104) | α n | cos (cid:16) k n ( x − x n ) (cid:17) e σ n t + iφ n + | β n | cos (cid:16) k n ( x − ˜ x n ) (cid:17) e − σ n t − iφ n (cid:105)(cid:111) + O ( (cid:15) ) , (17) where x n , ˜ x n are defined in (14) and, for | t − T (1) n | ≤ O (1) , we are in the firstnonlinear stage of MI, describing the first appearance of the AW through theformula u ( x, t ) = e iφ n F (cid:16) x, t ; φ n , X (1) n , T (1) n (cid:17) + O ( (cid:15) ) , (18) where function F is the Akhmediev breather: F ( x, t ; θ, X, T ) ≡ e it cosh[ σ ( θ )( t − T )+2 iθ ]+sin θ cos[ k ( θ )( x − X )]cosh[ σ ( θ )( t − T )] − sin θ cos[ k ( θ )( x − X )] ,k ( θ ) = 2 cos θ, σ ( θ ) = k ( θ ) (cid:112) − k ( θ ) = 2 sin(2 θ ) , (19)8 xact solution of focusing NLS for all real values of the parameters θ, X, T .The subsequent evolution is completely fixed by the recurrence property of thesolution u ( x + ∆ X n , t + ∆ T n ) = e i ∆ T +4 iφ n u ( x, t ) + O ( (cid:15) ) , x ∈ [0 , L ] , t ≥ . (20) Remark 3
We first observe that the first linear and nonlinear stages of MIdo match in the intermediate region O (1) (cid:28) t (cid:28) T (1) n = O ( σ − n | log (cid:15) | ) . Wealso remark that the periodicity property (20) implies that the solution de-scribes an exact recurrence of AWs, and the m th AW of the sequence ( m ≥ )is described, in the time interval | t − T ( m ) n | ≤ O (1) , by the analytic determin-istic formula u ( x, t ) = e i Φ ( m ) n F (cid:16) x, t ; φ n , X ( m ) n , T ( m ) n (cid:17) + O ( (cid:15) ) , m ≥ . (21)Therefore we have the following simple picture. The solution of the x -periodic Cauchy problem (3)-(7) describes, in the casein which the initial condition (15) opens only the pair of gaps associated with ± k n , an exact recurrence of Akhmediev breathers, whose parameters, chang-ing at each appearance, are expressed in terms of the initial data via elemen-tary functions. T (1) n is the first appearance time of the AW (the time at whichthe AW achieves the maximum of its modulus), X (1) n + Lj/n , ≤ j ≤ n − are the positions of such a maximum, φ n is the value of the maxi-mum, ∆ T n is the recurrence time (the time interval between two consecutiveAW appearances), ∆ X n is the x -shift of the position of the maxima in the re-currence. In addition, after each appearance, the AW changes the backgroundby the multiplicative phase factor exp(4 iφ n ) (see Figure 1). Remark 4 On the physical relevance of the above exact recurrenceof AWs
It is very important to remark that, if the number of unstable modesis greater than one (
N > ), this uniform in t dynamics is sensibly affected byperturbations due to numerics and/or real experiments. Indeed these pertur-bations open generically other small unstable gaps, provoking O (1) correctionsto the result. Therefore this analytic and uniform in t result is physically rel-evant (and numerically verifiable) only when N = n = 1 ( π < L < π ). x Figure 1: The level plot of | u ( x, t ) | , x ∈ [ − L/ , L/ L = 6 (the case of oneunstable mode k ), with c = 1 / , c − = (0 . − . i ) / , (cid:15) = 10 − . Thenumerical output is in perfect qualitative and quantitative agreement withthe theoretical formulas (16)-(21).2) In the second case, in which more than one unstable gap is opened bythe initial condition (15), a detailed investigation of all these gaps is necessaryto get a uniform in t dynamics, and this study is postponed to a subsequentpaper. It was however possible to obtain in [48] the elementary description ofthe first nonlinear stage of MI, described again by the Akhmediev breathersolution, and how perturbations due to numerics and/or real experimentscan affect this result. Remark 5 AW recurrence as basic effect of nonlinear MI in theperiodic setting, and the finite gap method
The recurrence of AWscan be predicted from simple qualitative considerations. The unstable modegrows exponentially and becomes O (1) at logarithmically large times, whenone enters the nonlinear stage of MI, and one expects the generation of atransient, O (1) , coherent structure over the unstable background (the AW).Since the Akhmediev breather describes the one-mode nonlinear instability, itis the natural candidate to describe such a stage, at the leading order. Due gain to MI, this AW is expected to be destroyed in a finite time interval, andone enters the third asymptotic stage, characterized, like the first one, by thebackground plus an O ( (cid:15) ) perturbation. This second linear stage is expected,due again to MI, to give rise to the formation of a second AW (the secondnonlinear stage of MI), described again by the Akhmediev breather, but, ingeneral, with different parameters. And this procedure iterates forever, in theintegrable NLS model, giving rise to the generation of an infinite sequence ofAWs described by different Akhmediev breathers. Then the AW recurrence isa relevant effect of nonlinear MI in the periodic setting, and the finite gapmethod is the proper tool to give an analytic description of it. Remark 6 Finite gap approach vs matched asymptotic expansions.
The above AW recurrence is described by an alternating sequence of linearand nonlinear asymptotic stages of MI, obviously matching in their overlap-ping time regions; therefore this finite gap result naturally motivates the in-troduction of a matched asymptotic expansions (MAEs) approach, presentedin the paper [49], and involving more elementary mathematical tools. Theadvantages of the finite gap approach are due to the fact that the θ -functionrepresentation of the solution is uniform in time, and the analytic descriptionof the nonlinear stages of MI (of the sequence of AWs) does not require anyguess work. Such a guess work is instead needed in the MAE approach, whenone has to select the proper nonlinear coherent structure of NLS describing acertain nonlinear stage of MI and matching with the preceeding linear stage.In all the situations in which such a guess work is no problem, the MAE ap-proach becomes competitive, since it involves more elementary mathematics,like in the case of one unstable mode only. But, as we shall see in the follow-ing sections, if we have more than one unstable mode, and the initial dataare generic, the dynamics is not described by a sequence of asymptotic stagesof MI, and the MAE approach is not applicable, while the finite gap methodis able to provide the uniform representation of the solution. The MAE ap-proach works, in the case of a finite number N > of unstable modes, onlyfor very special initial data [49]. Remark 7 AW recurrence as stable output of the dynamics.
Theo-rem 1 explains the relevance of the Akhmediev breather in a generic periodicCauchy problem of AWs, in the case of one unstable mode, and leads to thefollowing natural question: what happens if the initial condition is the highlynon generic Akhmediev breather? The answer is the following. The Akhme-diev breather, quasi homoclinic solution of NLS, is unstable, corresponding, n the finite gap spectral picture, to the highly non generic case in which allthe spectral gaps are closed. Any small perturbation (due, for instance, to thenumerical scheme approximating NLS, or to small corrections to NLS com-ing from physics) opens small gaps, implying that, after the first appearanceof the AW as predicted by the initial datum, a recurrence of AWs, describedby (16)-(21), will be the stable output of the dynamics [50]. Remark 8 AW recurrence for other NLS type models.
It is naturalto ask if the above AW recurrence is typical of NLS, or it is shared by otherintegrable NLS type models. In the case of the focusing Ablowitz - Ladikmodel [3] iu nt + u n +1 + u n − − u n + | u n | ( u n +1 + u n − ) = 0 , n ∈ Z , in-tegrable discretization of focusing NLS, the picture is essentially the same.Indeed, using MAEs and in the case of one unstable mode only, it was shownin [33] that a generic periodic initial datum leads to an AW recurrence ofNarita solutions (the Narita solution is the discrete analogue of the Akhme-diev breather [81, 16]). In the case of the PT-symmetric NLS (PT-NLS)equation [4, 5, 6, 7] iw t ( x, t ) + w xx ( x, t ) + 2 w ( x, t ) w ( − x, t ) = 0 , w = w ( x, t ) ∈ C , there are instead novelties with respect to the NLS case. Theanalogue of the Akhmediev breather, always regular for all values of its pa-rameters, is now a pair of exact solutions of PT-NLS [91]. The first one issingular in a certain domain of its parameter space, while the second one isalways singular, for all values of its arbitrary parameters; these singularitiesdescribe blow ups in points of the ( x, t ) plane. It follows that, depending onthe initial data, the AWs are either regular, with arbitrarily large amplitude,or they blow up at finite time [92]. This is a novel phenomenon for integrableNLS type systems, and a qualitative reason for it should be the gain-loss prop-erties of the complex self-induced potential of PT-NLS, causing extra-focusingeffects with respect to the NLS case. Remark 9 NLS exact recurrence vs Fermi-Pasta-Ulam recurrencein nature
From [48, 49] and formulas (16)-(21) it follows that, if the initialcondition (5) excites the only unstable mode k : u ( x,
0) = 1+ (cid:15) ( c exp( ik x )+ c − exp( − ik x )) , then: i) the energy is initially concentrated on the zero mode(the background) and on the first mode (the monochromatic perturbation): | u (0) | = 1 , | u m (0) | = δ m, ± (cid:15) | c ± | , (22) where u m ( t ) , m ∈ Z are the Fourier coefficients of the NLS solution u ( x, t ) .ii) At the first appearance time T (1)1 of the AW (see (16)), the energy is istributed on all Fourier modes according to the simple law | u ( T (1)1 ) | = (2 cos φ − , < φ < π , | u m ( T (1)1 ) | = 4(cos φ ) (cid:0) tan (cid:0) φ (cid:1)(cid:1) | m | , m (cid:54) = 0 , (23) but, iii) at the recurrence time ∆ T , it is re-absorbed by the zero and firstmodes: | u (∆ T ) | = 1 , | u m (∆ T ) | = δ m, ± (cid:15) | c ± | , (24) starting an exact recurrence [52].AW recurrence in the periodic setting has been already considered in theliterature (see, f.i., [105, 73, 106, 14, 103, 72]), and recent experiments inwater waves [59], in fiber optics [80], and in the nonlinear optics of a pho-torefractive crystal [87] accurately reproduce the recurrence phenomena. Inparticular, the experimental findings obtained in [87] have been compared withthe formulas (16)-(21) describing the NLS recurrence, obtaining a very goodqualitative and quantitative agreement. 1) Since NLS describes the abovedifferent physics only at the leading order, one expects that the exact NLS re-currence of AWs, illustrated in formulas (16)-(21), be replaced by a “Fermi-Pasta-Ulam” - type recurrence [46], before thermalization destroys the pat-tern. Indeed, in [87], up to 3 recurrences were observed and compared with theabove NLS recurrence formulas, obtaining a very good qualitative and quan-titative agreement. It was also shown that the recurrent behavior disappearswhen the photorefractive crystal works, instead, in a nonlinear regime differ-ent from the integrable (Kerr) one. 2) A common feature of the experiments[59, 80, 87] is the choice to work, in most of the cases, in a special symmetryof the experimental apparatus leading to the particularly significant subcasesin which the recurrence shift ∆ X in (16) is either or L/ , implying respec-tively the time-periods ∆ T or T , where ∆ T is the recurrence time (16).This particular symmetry corresponds to the distinguished sub-case in which | c | ∼ | c − | in (6), and leads to interesting physical resonances of the physicaltimes T (1)1 and ∆ T as functions of ζ = arg c +arg c − , for ζ = − φ , π − φ [51]; these resonances have been experimentally observed in [87]. The aboveexperimental findings, in good quantitative agreement with the theoretical for-mulas (16)-(21), are an important confirmation that NLS is a good model inthe description of nonlinear modulation instabilities in nonlinear optics andwater waves. In this paper we apply the finite gap method to the generic periodic13auchy problem (3)-(7) of the AWs, in the case of a finite number N of un-stable modes. Qualitative considerations similar to those made in Remark5 suggest again that each unstable mode will appear recurrently in the dy-namics, depending on its degree of instability; but this recurrence now isaffected by the nonlinear interactions with all the other unstable modes (weshall see that, at the leading order, this interaction is pairwise). Again theproper tool to describe all that is the finite gap method, and again the finitegap method adapts to this specific Cauchy problem through the three basicsimplifications outlined above, allowing one to construct the solution, at theleading and relevant order, in terms of elementary functions of the initialdata also in this more complicated case.More precisely, we will show that, at the leading order, i) the initial datagenerate a partition of the time axis into a sequence of finite intervals; ii) ineach interval I of the partition, only a subset of N ( I ) ≤ N unstable modes are“visible”, and iii) the NLS solution is approximated, for t ∈ I , by the N ( I )-soliton solution of Akhmediev type, describing the nonlinear interaction ofthese visible unstable modes, whose parameters are expressed in terms of theinitial data through elementary functions. This result explains the relevanceof the m -soliton solutions of Akhmediev type, with m ≤ N , in the genericperiodic Cauchy problem of the AWs, and in the case of a finite number N of unstable modes. Therefore, in the case of a finite number of unstablemodes, the theory of NLS anomalous waves is completely deterministic, andits analytic description, at the leading and relevant order, is given in termsof elementary functions of the Cauchy data .The paper is organized as follows. In Section 2 we describe the main re-sults. In Section 3 we summarize the classical features of the periodic Cauchyproblem for the focusing and defocusing NLS equations. In Section 4 we ap-ply this theory to the Cauchy problem of the anomalous waves, constructingthe main and auxiliary spectra at the leading and relevant order, through ele-mentary functions of the initial data. In Section 5, after closing the infinitelymany gaps corresponding to the stable modes, obtaining a non standard fi-nite gap approximation (the first basic simplification of the theory), we studythe corresponding finite gap curve, constructing the leading order expressionof the Riemann matrix, of the vector of Riemann constants, and of all theother quantities appearing in the parameters of the θ -functional formulas ofthe inverse problem, in terms of elementary functions (this is the second basicsimplification of the theory). In Section 6 we write the θ -functional formulasfor the leading order solution, and Section 7 is devoted to the presentation14f the third basic simplification of this theory, in which the infinite sum ofexponentials appearing in the definition of the θ function, is reduced to asum over a finite subset of exponentials, different in different time intervals.This paper will appear in Novikov’s volume, on the occasion of his 80thbirthday. In this respect, it is worth mentioning that1) Novikov has always been interested in applying his results to realphysics.2) Novikov has always pointed out that the finite-gap formulas requireadditional effectivization to become applicable.We hope we made here a serious progress in both directions. This paperis dedicated to S.P. Novikov. The aim of this paper is to provide the solution, at the leading order and interms of elementary functions, of the generic periodic Cauchy problem of theAWs (3)-(7) described in the Introduction, rewritten here for completeness: iu t + u xx + 2 | u | u = 0 , u = u ( x, t ) ∈ C , x ∈ [0 , L ] , t ≥ ,u ( x + L, t ) = u ( x, t ) ,u ( x,
0) = 1 + (cid:15)v ( x ) , | (cid:15) | (cid:28) ,v ( x ) = (cid:80) j ≥ (cid:0) c j e ik j x + c − j e − ik j x (cid:1) , k j = πL j, | c j | = O (1) , (25)where the period L is generic ( L/π is not an integer).We first list the ingredients we need to construct its solution.1. The number N of the unstable modes N = (cid:22) Lπ (cid:23) ; (26)2. their wave numbers k j and growth rates σ j : k j = πL j, σ j = k j (cid:113) − k j , ≤ j ≤ N, (27)and the angles φ j parametrizing them: φ j = arccos (cid:18) k j j (cid:19) = arccos (cid:16) πL j (cid:17) , < φ j < π , ≤ j ≤ N, (28)15o that k j = 2 cos( φ j ) , σ j = 2 sin(2 φ j ) . (29)We also define the 2 N angles ˆ φ j by:ˆ φ j = φ j , ˆ φ j + N = − φ j , j = 1 , . . . , N. (30)3. The following linear combinations of the Fourier coefficients of the unstablemodes: α j = ( e − iφ j c j − e iφ j c − j ) , β j = ( e iφ j c − j − e − iφ j c j ) , j = 1 , . . . , N ; (31)and the quantitiesˆ α j = α j , ˆ α j + N = β j , ˆ β j = β j , β j + N = α j , j = 1 , . . . , N. (32)4. The leading order 2 N × N Riemann matrix B = ( b jk ): b jj = 2 log (cid:15) (cid:113) ˆ α j ˆ β j | φ j ) cos( ˆ φ j ) | , j = 1 , . . . , N, (33) b jk = 2 log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin (cid:16) ˆ φ j − ˆ φ k (cid:17) cos (cid:16) ˆ φ j + ˆ φ k (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , j (cid:54) = k, j, k = 1 , . . . , N, (34)assuming that Re (cid:113) ˆ α j ˆ β j ≥ , j = 1 , . . . , N, (cid:113) ˆ α j + N ˆ β j + N = (cid:113) ˆ α j ˆ β j , j = 1 , . . . , N.
5. The following 2 N dimensional complex vectors: (cid:0) (cid:126)z − ( x, t ) (cid:1) j = iπ − log ˆ α j (cid:113) ˆ α j ˆ β j + 2 i cos( ˆ φ j ) x − φ j ) t, (35) (cid:0) (cid:126)z + ( x, t ) (cid:1) j = (cid:0) (cid:126)z − ( x, t ) (cid:1) j − πi − i ˆ φ j , j = 1 , . . . , N, (cid:126)z + ( x, t )) j = Re( (cid:126)z − ( x, t )) j = − φ j ) t − log (cid:12)(cid:12)(cid:12) ˆ α j ˆ β j (cid:12)(cid:12)(cid:12) , Re( (cid:126)z ± ( x, t )) j + N = − Re( (cid:126)z ± ( x, t )) j , j = 1 , . . . , N, (36)and these expressions do not depend on x .6. The 2 N -dimensional real vector: (cid:126)w ( t ) = (Re B ) − Re (cid:126)z − ( x, t ) = (Re B ) − Re (cid:126)z + ( x, t ) , (37)describing the corresponding straight line time evolution in R N , equippedwith the standard integer lattice Z N . From (31), (34), (36), it follows im-mediately that w j + N ( t ) ≡ − w j ( t ) , j = 1 , . . . , N. (38)Equation (38) suggests that the construction of the relevant components of (cid:126)w ( t ) can be simplified, involving the inversion of a simpler to handle N × N matrix. Indeed, if we introduce the N-vector w ( t ) ∈ R N whose componentsare the first N components of (cid:126)w ( t ), then: (cid:126)w ( t ) = ( w ( t ) , − w ( t )) T ∈ R N ,w ( t ) = w (1) t + w (0) ,w (1) = −B − σ, w (0) = −B − χ, (39)where B − is the inverse of the real symmetric N × N matrix B defined by B jj = − σ j τ j , B jk = 2 log (cid:12)(cid:12)(cid:12)(cid:12) sin( φ j − φ k )sin( φ j + φ k ) (cid:12)(cid:12)(cid:12)(cid:12) , ≤ j, k ≤ N, j (cid:54) = k, (40)with τ j = 2 σ j log (cid:32) σ j (cid:15) (cid:112) | α j β j | (cid:33) , ≤ j ≤ N, (41)and the N -vectors σ, χ ∈ R N are defined by σ = σ ... σ N , χ = log( (cid:112) | α /β | )...log( (cid:112) | α N /β N | ) . (42)17. The real number p , 0 < p ≤
1, characterizing the accuracy of the approxi-mation ˆ u ( x, t ) we want to achieve in the construction of the solution u ( x, t ): u ( x, t ) = ˆ u ( x, t ) + O ( (cid:15) p ). It follows that, if, at a given time, the contributionof a certain unstable mode to the solution is less than O ( (cid:15) p ), this mode isneglected, because invisible with respect to the above approximation. Definition 1
An unstable mode is “ p -visible” at a certain time t , if its con-tribution to the solution is of order (cid:15) p , or greater. It is “ p -invisible” otherwise.
8. The following 2 N vector (cid:126)st ( t ) ∈ R N of components st j ( t ) = 1 − Θ (cid:18) − w j ( t ) + (cid:98) w j ( t ) (cid:99) + 1 − p (cid:19) +Θ (cid:18) w j ( t ) − (cid:98) w j ( t ) (cid:99) − p (cid:19) , j = 1 , . . . , N, (43)where Θ( x ) in (43) denotes the standard step function:Θ( x ) = (cid:26) , x ≤ , , x > . Equivalently, st j ( t ) = w j ( t ) − (cid:98) w j ( t ) (cid:99) < − p , − p ≤ w j ( t ) − (cid:98) w j ( t ) (cid:99) ≤ p , w j ( t ) − (cid:98) w j ( t ) (cid:99) > p . with: st j + N ( t ) + st j ( t ) = 2 , for all t and j = 1 , . . . , N. The vector (43) is used to detect which unstable mode is “ p -visible” at agiven time. The j -th mode is “ p -visible” at time t , iff st j ( t ) = 1, and “ p -invisible” if st j ( t ) = 0 or st j ( t ) = 2 (see Figure 2).18 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id:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) Figure 2: The straight line trajectory w ( t ) ∈ R N for N = 2, with p = 1 / Theorem 2
Consider the generic periodic Cauchy problem of the anomalouswaves (25), for x ∈ [0 , L ] and t in any finite time interval [0 , T ] . Introducethe positive quantities t ( j ) k , j = 1 , . . . , N , k ≥ defined by t ( j )2 n − = − p + n − − w (0) j w (1) j , t ( j )2 n = p + n − − w (0) j w (1) j , n ≥ , ≤ j ≤ N, (44) where w (0) j , w (1) j , ≤ j ≤ N , are the N components of vectors w (0) , w (1) ,defined in (39)-(42). They are the times at which the j th unstable modechanges its status, from p-invisible to p-visible if k is odd, and from p-visibleto p-invisible if k is even, exhibiting the following recurrence properties t ( j )2 n − t ( j )2 n − = pw (1) j , t ( j )2 n +1 − t ( j )2 n = 1 − pw (1) j , (45) t ( j )2 n +1 − t ( j )2 n − = t ( j )2 n +2 − t ( j )2 n = 1 w (1) j , j = 1 , . . . , N, n ≥ . hey naturally partition the interval [0 , T ] in a sequence of finite intervals.In a given interval I of the partition, the j th unstable mode is p-visible iff − p ≤ w j ( t ) − (cid:98) w j ( t ) (cid:99) ≤ p , ≤ j ≤ N, t ∈ I, (46) and p-invisible otherwise. Let us denote by N ( I ) the number of “ p -visible”modes in the interval I , and by s k ( I ) , k = 1 , . . . , N ( I ) the indices of these“ p -visible” modes, assuming that s k + N ( I ) ( I ) = s k ( I ) + N . Then the solutionof the Cauchy problem, at the leading and relevant order, is described by thefollowing formula u ( x, t ) = u I ( x, t ) + O ( (cid:15) p ) , t ∈ I, (47) where u I ( x, t ) is the exact N ( I ) -soliton solution of Akhmediev type describingthe nonlinear interaction of the N ( I ) unstable modes that are visible in theinterval I , L -periodic in x and localized in time over the background: u I ( x, t ) = exp(2 i Φ( I )) · A N ( I ) ( x, t ) , defined by A N ( I ) ( x, t ) = exp(2 it ) ˆ θ N ( I ) ( ˆ Z + ( x, t ) | B )ˆ θ N ( I ) ( ˆ Z − ( x, t ) | B ) , (48) where (we keep only the “p-visible” components in the sum): ˆ θ N ( I ) ( ˆ Z | B ) = (cid:88) ˆ n j ∈ {− , } j = 1 , . . . , N ( I )exp N ( I ) (cid:88) l, m = 1 l (cid:54) = m log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin (cid:18) ˆ φ sl ( I ) − ˆ φ sm ( I ) (cid:19) cos (cid:18) ˆ φ sl ( I ) + ˆ φ sm ( I ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ n l ˆ n m πi N ( I ) (cid:88) l =1 ˆ n l ˆ Z l , (49) (cid:16) ˆ Z − ( x, t ) (cid:17) j = (ˆ z − ( x, t )) s j ( I ) , j = 1 , . . . , N ( I ) , , ( ˆ Z + ( x, t )) k = ( ˆ Z − ( x, t )) k − − ˆ φ s k ( I ) π , ˆ z − ( x, t ) = (cid:126)z − ( x, t ) − N (cid:88) k =1 (cid:20) (cid:98) w k ( t ) (cid:99) + st k ( I )2 (cid:21) (cid:126)b k , I ) = N (cid:88) k =1 [2 (cid:98) w k ( t ) (cid:99) + st k ( I )] ˆ φ k , (50) and (cid:126)b k is the k th column of matrix B . Therefore, in each time interval I of the partition, we approximate u ( x, t ), upto O ( (cid:15) p ), 0 < p ≤ L = 10), and p = 1 /
2, in very good agrement with the correspondingnumerical experiment. 21 t (3)4 t (3) 1 t (2)4 t (2) 3 t (2) 2 t (2)3 t (3)5 t (2)7 t (2) 1 t (3)1 t (1)2 t (1)3 t (1)6 t (2)4 t (1)5 t (1) Figure 3: The graph of | u ( x, t ) | at the leading order, from formulas (47)-(50),in good agreement with the corresponding numerical experiment. Here L =10 (N=3), 0 ≤ t ≤ x ∈ [ − L/ , L/ , (cid:15) = 10 − , c = 0 . c − = 0 . . i , c = 0 . c − = − .
03 + 0 . i , c = 0 . c − = 0 .
02 + 0 . i , p = 1 /
2, andthe short axis is the x -axis. The intervals of “p-visibility” for the 3 unstablemodes are marked, below the graph, by bold lines. The boundary points ofthe partition intervals are, sequentially, 0, t (2)1 , t (1)1 , t (3)1 , t (2)2 , t (1)2 , t (2)3 , t (3)2 , t (2)4 , t (1)3 , t (3)3 , t (2)5 , t (1)4 , t (2)6 , t (3)4 , t (2)7 , t (1)5 . In the first interval (0 , t (2)1 ) we are in thefirst linear stage of MI, and all modes are invisible; in the interval ( t (2)1 , t (1)1 )the most unstable mode 2 is visible; in the interval ( t (1)1 , t (3)1 ) also the mode1 is visible; in the interval ( t (3)1 , t (2)2 ) all the three modes are visible; andso on. The unstable mode 2, characterized by two maxima in the period,is the most unstable mode ( σ = max { σ , σ , σ } ). It first appears in theinterval ( t (2)1 , t (2)2 ); it desappears in the interval ( t (2)2 , t (2)3 ), reappearing againin the interval ( t (2)3 , t (2)4 ); and so on. Its recurrence properties are ruled byequations (45). Remark 10
To the best of our knowledge, the exact N -soliton solution ofAkhmediev type has the following history. The representation (48) was ob-tained by Its, Rybin and Sall in [62], as a degeneration of the finite-gapformulas. The determinant form of this solution and a discussion on theconnection between these two representations are also provided in [62]. Letus remark that formulas (48) can be obtained from the Hirota N -soliton solu- ion [56] of defocusing NLS by a complex rotation [29]. Determinant formu-las for the N -soliton solution over the zero background for the self-focusingcase were provided in the book of Faddeev and Takhtadjan [43], and the de-terminant formula for the N -soliton solution over an arbitrary background,including the so-called super-regular solitons, was constructed by Zakharovand Gelash [108]; see also [109], [110]. Remark 11
Formula (49) involves summations over N , N ≤ N termsof O (1) , growing exponentially with N (f.i., if N = 5 , the sum involvesabout terms). It follows that these sums could generate contributionscomparable with O ( | log (cid:15) | ) , for reasonable values of (cid:15) coming from physicsor from numerical simulations, and the results would become less reliable.Therefore the number N of unstable modes must be sufficiently small to avoidthis problem. Remark 12
In this paper we use a more symmetric notation with respect tothe one used in our previous works on this subject [48]-[52] (see, f.i., equations(31). We also use a different normalization for the theta-functions: in thepresent text it coincides with the one used in [44], [19], [39], while, in theprevious works, we used the normalization from the book [79].
Let us recall the basic facts about the periodic theory of the NonlinearSchr¨odinger (NLS) equation.The NLS equation has two real forms: iu t + u xx − | u | u = 0 , u = u ( x, t ) ∈ C , defocusing NLS iu t + u xx + 2 | u | u = 0 , u = u ( x, t ) ∈ C , self-focusing NLS . (51)In nonlinear optics, both models describe media with refractive index de-pending on the electric filed. If the refractive index decreases (increases) inthe presence of the electromagnetic wave, we have defocusing (respectivelyself-focusing) NLS equation. Both forms can be treated as real reductions ofthe complex NLS equation (cid:26) iq t + q xx + 2 q r = 0 , − ir t + r xx + 2 qr = 0 , q = q ( x, t ) , r = r ( x, t ) ∈ C , (52)23here q ( x, t ) = u ( x, t ) , r ( x, t ) = − u ( x, t ) defocusing case (53) q ( x, t ) = u ( x, t ) , r ( x, t ) = u ( x, t ) self-defocusing case (54)The integration of the NLS equation is based on the zero-curvature repre-sentation found by Zakharov and Shabat in [114]. A pair of functions q ( x, t ), r ( x, t ) satisfies the complex NLS equation (52) if and only if the followingpair of linear problems is compatible: (cid:126) Ψ x ( λ, x, t ) = U ( λ, x, t ) (cid:126) Ψ( λ, x, t ) , (55) (cid:126) Ψ t ( λ, x, t ) = V ( λ, x, t ) (cid:126) Ψ( λ, x, t ) , (56) U = (cid:34) − iλ iq ( x, t ) ir ( x, t ) iλ (cid:35) , where V ( λ, x, t ) = (cid:34) − iλ + iq ( x, t ) r ( x, t ) 2 iλq ( x, t ) − q x ( x, t )2 iλr ( x, t ) + r x ( x, t ) 2 iλ − iq ( x, t ) r ( x, t ) (cid:35) , and (cid:126) Ψ( λ, x, t ) = (cid:20) Ψ ( λ, x, t )Ψ ( λ, x, t ) (cid:21) . The first equation of the zero-curvature representation (55) can be rewrittenas the following spectral problem L (cid:126) Ψ( λ, x, t ) = λ(cid:126) Ψ( λ, x, t ) , (57)where L = (cid:20) i∂ x q ( x, t ) − r ( x, t ) − i∂ x (cid:21) . The main tool for constructing periodic and quasiperiodic solutions of solitonsystems is the finite-gap method, invented by S.P. Novikov [82] for the pe-riodic Korteweg - de Vries (KdV) problem. Finite-gap solutions of the NLSequation were first constructed by Its and Kotljarov [61].Let us recall the principal facts about the periodic direct and inversespectral transform for the 1-dimensional Dirac operator (57).24onsider a fixed time t = t . Let q ( x ), r ( x ) denote the Cauchy data q ( x ) = q ( x, t ), r ( x ) = r ( x, t ), q ( x + L ) = q ( x ), r ( x + L ) = r ( x ). The directspectral transform associates to these Cauchy data the following spectraldata:1. The spectral curve Γ, i.e., the Riemann surface for the Bloch eigen-functions.2. The divisor, i.e., the set of eigenvalues for an auxiliary Dirichlet-typespectral problem.
1. The spectral curve.
The Bloch functions of L are defined as thecommon eigenfunctions of L and of the translation operator (the periodic-ity of L means that it commutes with the translation by the basic period L ; therefore L and the translation operator have sufficiently many commoneigenfunctions): L (cid:126) Ψ( x, t ) = λ(cid:126) Ψ( x, t ) ,(cid:126) Ψ( x + L, t ) = κ(cid:126) Ψ( x, t ) , γ ∈ Γ . (58)Equivalently, the Bloch functions are eigenfunctions of the monodromy ma-trix ˆ T ( λ, x , t ) defined by:ˆ T ( λ, x , t ) = ˆΨ( λ, x + L, t ) , where the 2 × λ, x, t ) denotes the solution of the matrixequation L ˆΨ( λ, x, t ) = λ ˆΨ( λ, x, t ) , with the following initial conditionˆΨ( λ, x , t ) = (cid:20) (cid:21) . The monodromy matrix ˆ T ( λ, x , t ) is holomorphic in λ in the whole complexplane. It follows that the eigenvalues and eigenvectors of ˆ T ( λ, x , t ) (i.e.,the Bloch multiplier κ and the Bloch eigenfunction (cid:126) Ψ( x, t )) are defined ona two-sheeted covering Γ( x , t ) of the λ -plane; therefore we shall write κ ( γ ), (cid:126) Ψ( γ, x, t ), γ ∈ Γ.The Riemann surface Γ( x , t ) is called the spectral curve . If the po-tentials q ( x, t ), r ( x, t ) satisfy the NLS equation (52), then the monodromy25atrices corresponding to different x and t coincide up to conjugation;therefore Γ( x , t ) does not depend on x and t , and will be denoted by Γin the remaining part of the text. In particular, this means that the curve Γprovides an infinite set of conservation laws for the NLS hierarchy. This setis complete, and the standard local conservation laws can be easily obtainedas expansion coefficients of Γ near infinity.The spectrum of the problem (57) in L ( R ) is always continuous, and itis defined by the following property: λ ∈ C belongs to the spectrum of L if and only if equation (57) admits a solution growing not faster then somepolynomial for both x → −∞ and x → + ∞ . The ends of the spectralintervals coincide with the branch points of Γ. In the paper [82] the spectralcurve was introduced for the real Schr¨odinger operator, where the relationbetween the spectral curve and the classical spectrum is especially simple:the spectrum in L ( R ) is the set of intervals in the real line bounded by thebranch points of Γ.We use the following notation: if γ ∈ Γ is a point of our spectral curve,then λ ( γ ) denotes the projection of the point γ to the λ -plane.The multivalued function p ( γ ) = 1 iL log( κ ( γ )) (59)is called quasimomentum. Its differential dp ( γ ) is well-defined and mero-morphic on Γ with two simple poles at the infinity points, and all periods of dp are real. The trace of matrix U ( λ, x, t ) is equal to zero; thereforedet ˆ T ( λ, x , t ) ≡ , and if λ ( γ ) = λ ( γ ), then κ ( γ ) κ ( γ ) = 1.The “classical” spectrum of L in L ( R ) coincides with the projection ofthe set { γ ∈ Γ , | κ ( γ ) | = 1 } to the λ -plane, or, equivalently, is defined by thecondition: Im p ( γ ) = 0 . The branch points of Γ coincide with the ends of the spectral zones.
Definition 2
A pair of potentials q ( x ) , r ( x ) is called finite-gap if the spec-tral curve Γ is algebraic, i.e., it has only a finite number of branch pointsand non-removable double points. In the real case r ( x ) = ± q ( x ) , the secondrequirement is fulfilled automatically, and it is sufficient to demand that thenumber of branch points be finite. emark 13 The analytic properties of the Bloch eigenfunction for the 1-dimensional Schr¨odinger operator in the domain of complex energies werefirst studied in the paper by Kohn [67].
2. Divisor
The auxiliary spectrum is defined as the set of points γ ∈ Γsuch that the first component of the Bloch eigenfunction is equal to 0 at thepoint x . L (cid:126) Ψ( γ, x, t ) = λ ( γ ) (cid:126) Ψ( γ, x, t ) ,(cid:126) Ψ( γ, x + L, t )) = κ ( γ ) (cid:126) Ψ( γ, x, t ) , (60)Ψ ( γ, x , t ) = 0 . Equivalently, the auxiliary spectrum coincides with the set of zeroes of thefirst component of the Bloch eigenfunction:Ψ ( γ, x, t ) = 0; (61)therefore it is called the divisor of zeroes. The zeroes of Ψ ( γ, x, t ) dependon x and t . The x , t dynamics becomes linear after the Abel transform (forKdV this fact was first established by Dubrovin [38], Its and Matveev [60]). Remark 14
Let us point out that, in some papers dedicated to the finite-gapNLS solutions, a different auxiliary problem is used. Namely, one imposesthe following symmetric boundary condition: Ψ ( λ, x , t ) + Ψ ( λ, x , t ) = Ψ ( λ, x + L, t ) + Ψ ( λ, x + L, t ) = 0 . (62) This approach has the following advantage: all divisor points are located in acompact area of the spectral curve, but it requires one extra divisor point andincreases the complexity of the formulas.
The wave propagation in focusing and defocusing media is substantiallydifferent from the physical point of view. This difference means that theanalytic properties of the solutions are also substantially different (this topicis well-presented in the paper by Previato [89]).1. In the defocusing case: • Operator L is self-adjoint, and its spectrum is real; • All branch points of Γ are real and simple;27
If the normalization (62) is used, each non-empty spectral gapcontains exactly one divisor point; • The defocusing NLS equation possesses regular and singular finite-gap solutions.From the analytic point of view, the theory of defocusing NLS is anal-ogous to the theory of the real Korteweg-de Vries equation.2. In the self-focusing case: • Operator L is non-self-adjoint, and the matrix U ( λ, x, t ) is skew-hermitian for real λ . Therefore, for each real λ , the monodromymatrix is unitary. It means that the whole real line lies in the L ( R ) spectrum of L ; in addition, generically, the L ( R ) spectrumof L contains some arcs in the complex domain. • All branch points of Γ are complex, and a finite number of themmay be of high odd multiplicity. All real double points of Γ areremovable (the monodromy matrix has two different eigenfunc-tions), but complex double points may be non-removable, andsuch points are associated with homoclinic orbits. For sufficientlyregular data, the number of complex double points is always fi-nite. The curve Γ is real; i.e., it is invariant with respect to thecomplex conjugation. • The characterization of the divisor is less explicit, and it can beprovided in terms of Cherednik differentials [32]. • All finite-gap solutions are automatically regular [32].In contrast with the defocusing case, the self-focusing case is muchricher and much more interesting from the analytic point of view.For generic smooth NLS Cauchy data q ( x, t ), r ( x, t ), the surface Γ hasinfinite genus, but for large values of the spectral parameter, the branchpoints of Γ form close pairs (see [36] for analytic estimates extending theresults of [57] to the 1-d Dirac operator), and one can construct an arbitrarilygood finite-gap approximation for a given smooth potentials by merging allclose pairs of branch points except a finite number. “Naive” merging does notrespect the spatial periodicity and provides a local in x approximation only.In the defocusing case, a period preserving approximation may be obtained28sing a minor modification of the Marchenko-Ostrovskii approach [76]. Inthe focusing case, the analytic properties of the quasimomentum are morecomplicated, and, to construct a pure periodic finite-gap approximation, onecan use either the isoperiodic deformation (Grinevich-Schmidt [53]), or aproper adaptation of the Krichever’s technique from the paper [69].Therefore, in the periodic problem for soliton equations, the role of finite-gap potentials can be compared with that of finite Fourier series in the theoryof linear PDEs.The solution of the inverse problem in the finite-gap case (the genus of Γis equal to g < ∞ ) is provided by the theta-functional formula ([61], see also[89]): u ( x, t ) = C exp( U x + V t ) θ ( (cid:126)A ( ∞ − ) − (cid:126)U x − (cid:126)U t − (cid:126)A ( D ) − (cid:126)K | B ) θ ( (cid:126)A ( ∞ + ) − (cid:126)U x − (cid:126)U t − (cid:126)A ( D ) − (cid:126)K | B ) , (63)where C , U , V are constants defined in terms of the spectral curve, (cid:126)A ( D ), (cid:126)A ( ∞ + ), (cid:126)A ( ∞ − ) are the Abel transforms of the divisor, and of the infinitypoints of Γ respectively, (cid:126)K is the so-called vector of Riemann constants, B isthe Riemann period matrix for Γ, (cid:126)U , (cid:126)U are the vectors of the b -periods forthe quasimomentun and quasienergy differentials, respectively, (see formulas(95), (99)), and θ ( z | B ) denotes the Riemann theta-function of genus gθ ( z | B ) = (cid:88) n j ∈ Z j = 1 , . . . , g exp (cid:34) g (cid:88) j,k =1 b jk n j n k + g (cid:88) j =1 n j z j (cid:35) . (64)where b jk are the components of matrix B . More informations about theta-functional formulas can be found in [39], [19]. Explicit approximate formulasfor the parameters appearing in (63),(64), in the special Cauchy problem ofanomalous waves, are provided in Section 4.The normalization (64) implies the following periodicity properties: θ ( (cid:126)z + (cid:126)a l | B ) = θ ( (cid:126)z | B ) θ (cid:16) (cid:126)z + (cid:126)b l | B (cid:17) = θ ( (cid:126)z | B ) exp (cid:18) − b ll − z l (cid:19) , (65)where l = 1 , . . . , N . 29 Spectral transform for the Cauchy problemof the anomalous waves
If one uses the focusing NLS equation as a mathematical model for anoma-lous waves, special initial data (5)-(6) are considered. We show that thepresence of a small parameter (cid:15) in this problem allows one to construct agood approximation for the solutions in terms of elementary functions.To construct the direct spectral transform for this problem, it is conve-nient to write L = L + (cid:15) L , L = (cid:20) i∂ x − − i∂ x (cid:21) , L = (cid:20) v ( x ) − v ( x ) 0 (cid:21) , and calculate the spectral data for L using the standard perturbation theorynear the spectral data for L . The leading order formulas for the spectralcurve and the divisor were provided in the recent paper [48] of the authors.Let us present a brief review of these results. The unperturbed spectral curve Γ for L is rational, and a point γ ∈ Γ is a pair of complex numbers γ = ( λ, µ ) satisfying the following quadraticequation: µ = λ + 1 . The Bloch eigenfunctions for the operator L can be easily calculated explic-itly: ψ ± ( γ, x ) = (cid:20) λ ( γ ) ± µ ( γ ) (cid:21) e ± iµ ( γ ) x , (66) L ψ ± ( γ, x ) = λ ( γ ) ψ ± ( γ, x ) . These eigenfunctions are periodic (antiperiodic) if and only if L π µ ∈ Z is aneven (an odd) integer. Let us introduce the following notation: µ n = πnL , λ n = (cid:112) µ n − , Re λ n + Im λ n > , n = 0 , , , . . . ∞ . Then the periodic and antiperiodic spectral points of L are (see Figure 5): {± λ n } , n = 0 , , , . . . ∞ .
30y analogy with [69], [70], these points are called resonant point , because,under generic small perturbations, they split into pairs of branch points. λ λ λ −λ −λ −λ λ λ −λ −λ Unstable resonantpointspointsStable resonant
Figure 4: The spectrum of the unperturbed operator L .We also use the following notation: λ − n = λ n , µ − n = − µ n , n ≥ . Let us recall that we assume L (cid:54) = πn , where n ∈ N + ; therefore the point λ = 0is not resonant. The case of resonant λ = 0 requires a special treatment, andwe plan to provide it in the future.We have the following basis of eigenfunctions for the periodic and an-tiperiodic problems: ψ ± n = (cid:20) µ n ± λ n (cid:21) e iµ n x , n ∈ Z , (67) L ψ ± n = ± λ n ψ ± n . The curve Γ has two branch points E = i , E = − i corresponding to n = 0. If n >
0, there is no branching at the resonant points ± λ n , but themonodromy matrix becomes diagonal with coinciding eigenvalues:ˆ T ( ± λ n , ,
0) = (cid:34) ( − n
00 ( − n (cid:35) . Resonant points are also the eigenvalues of the Dirichlet problem (60), i.e., the divisor points for the unperturbed problem , and the Dirichlet31igenfunctions are given by ψ Dir n ( x ) = ψ + n ( x ) − ψ + − n ( x ) , ψ Dir − n ( x ) = ψ − n ( x ) − ψ −− n ( x ) , n > . Taking into account that the squared eigenfunctions provide the properbasis for the linearized theory, we notice that the Fourier modes correspondthe the points µ n = k n /
2. These modes are unstable if the corresponding λ n are imaginary ( | µ n | <
1) and stable if the λ n are real ( | µ n | ≥ | n | < Lπ are unstable , andspectral points with | n | ≥ Lπ are stable . For the unstable modes we have: λ j = i cos( φ j ) , µ j = sin( φ j ) , j = 1 , . . . , N, (68)where the angles φ j are the same as in (11). To calculate the perturbed spectral curve, we develop the perturbation theoryfor the periodic and antiperiodic problems using the basis (67).It is also convenient to introduce the following notation where n ≥ α n = ( µ n − λ n ) c n − ( µ n + λ n ) c − n , β n = ( µ n + λ n ) c − n − ( µ n − λ n ) c n , (69)˜ α n = ( µ n + λ n ) c n − ( µ n − λ n ) c − n , ˜ β n = ( µ n − λ n ) c − n − ( µ n + λ n ) c n , (70)which is consistent with (31) since, for the unstable modes, e ± iφ n = µ n ± λ n .Let us introduce the following notation: | l ± > denotes the basic vector ψ ± l , < l ± | denote the adjoint basis: < l + | m + > = δ lm , < l − | m − > = δ lm , < l + | m − > = < l − | m + > = 0 . For an arbitrary periodic perturbation, the matrix elements of L can bewritten in the following form: < m + | L | l + > = c ( m − l ) / ( λ m − µ m )( λ l + µ l ) − c − ( m − l ) / λ m ,< m − | L | l + > = c ( m − l ) / ( λ m + µ m )( λ l + µ l ) + c − ( m − l ) / λ m ,< m + | L | l − > = c ( m − l ) / ( λ m − µ m )( − λ l + µ l ) − c − ( m − l ) / λ m , m − | L | l − > = c ( m − l ) / ( λ m + µ m )( − λ l + µ l ) + c − ( m − l ) / λ m . Here we use a slightly non-standard notation: < f | L | g > denotes the matrixelement for both orthogonal and non-orthogonal bases. We also assume that c − ( m − l ) / = 0, if m − l is odd.Using the standard perturbation theory (see [48] for details), one showsthat the resonant points λ n and − λ n generically split into the pairs of branchpoints { E n − , E n } and { ˜ E n − , ˜ E n } , where: E = i + O ( (cid:15) ) E l = λ n ∓ (cid:15) λ n (cid:112) α n β n + O ( (cid:15) ) , l = 2 n − , n, (71)˜ E l = − λ n ± (cid:15) λ n (cid:113) ˜ α n ˜ β n + O ( (cid:15) ) , l = 2 n − , n . (see Figure 6). Here we assume that Re √ α n β n ≥
0, Re (cid:113) ˜ α n ˜ β n ≥ √ α n β n <
0, Im (cid:113) ˜ α n ˜ β n < λ , we also used the constraint c = 0.For the perturbations of the unstable points we have:˜ E l = E l . (72)33 λ E E E λ λ λ E E E E E E E −λ −λ E E E E E E E E λ λ −λ −λ Figure 5: The spectrum of the perturbed operator L = L + (cid:15) L .Let us define the following enumeration for the unperturbed divisor points γ n = ( λ div n , µ div n ), n (cid:54) = 0: λ div n = (cid:26) λ n , n > , − λ n , n < , µ div n = µ n . The calculation of the divisor positions up to O ( (cid:15) ) corrections uses the sameperturbation theory with the following modification. In contrast with thebranch points, the Bloch multipliers for the Dirichlet spectrum are genericallydifferent from ±
1; moreover their absolute values do not have to be equal to1. Therefore we use the additional constraint that the first component of theDirichlet eigenfunction vanishes at the boundary of the interval to determinesimultaneously the variation of λ and the variation of the Bloch multiplier κ , and we obtain (see [48]): λ ( γ n ) = λ n + (cid:15) λ n [( µ n + λ n ) α n + ( µ n − λ n ) β n ] + O ( (cid:15) ) ,p ( γ n ) = (cid:15) µ n [( µ n + λ n ) α n − ( µ n − λ n ) β n ] + O ( (cid:15) ) , (73)34nd λ ( γ − n ) = − λ n − (cid:15) λ n (cid:104) ( µ n − λ n ) ˜ α n + ( µ n + λ n ) ˜ β n (cid:105) + O ( (cid:15) ) ,p ( γ − n ) = (cid:15) µ n (cid:104) ( µ n − λ n ) ˜ α n − ( µ n + λ n ) ˜ β n (cid:105) + O ( (cid:15) ) . (74) A generic small perturbation of the constant solution generates an infinitegenus spectral curve. Of course, it is natural to approximate it by a finite-genus curve. The perturbations corresponding to stable resonant points re-main of order (cid:15) for all t and can be well-described by the linear perturbationtheory. Therefore one can close all gaps associated with stable points, ob-taining a finite-gap approximation of the spectral curve. Remark 15
This finite-gap approximation is rather non-standard. Usually,one keeps the gaps associated with sufficiently large harmonics and close thegaps corresponding to the small ones; therefore the genus of the approximat-ing curve depends on the perturbation. In our problem, all harmonics of theperturbation are small, and the genus of the approximating curve is deter-mined by the number of unstable modes and does not depend on the Cauchydata.
Starting from this point, we shall use the following 2 N -gap approximationof the spectral curve: we open only the resonant points associated with theunstable modes, and do not perturb the stable double point.Our next step is to calculate the leading order approximation of thealgebro-geometrical data.Let us introduce the following notations:ˆ λ j = λ j , if 1 ≤ j ≤ N,λ j − N , if N + 1 ≤ j ≤ N, ˆ µ j = µ j , if 1 ≤ j ≤ N,µ j − N , if N + 1 ≤ j ≤ N. γ j = γ j if 1 ≤ j ≤ Nγ − j if N + 1 ≤ j ≤ N We have 4 N + 2 branch points: E j , j = 0 , . . . , N, E j , j = 0 , . . . , N, and the curve Γ is defined by: µ = N (cid:89) j =0 ( λ − E j )( λ − E j ) . (75)We also define µ (0) ( λ ) = √ λ + 1 ,z n = E n − + E n λ n + O (cid:0) (cid:15) (cid:1) . The natural compactification of Γ has two infinity points: ∞ + : µ ∼ − λ N +1 , ∞ − : µ ∼ λ N +1 . We have the following system of cuts (marked by the dashed lines in Figure 6):[ i ∞ , E ], [ E , E ],. . . , [ E N − , E N ], [ E N , E N − ], [ E , E ], [ E , − i ∞ ]. Weuse a slightly non-standard agreement: ∞ + is located in Sheet 2 (dashedlines), and ∞ − is located in Sheet 1 (solid lines).To obtain convenient formulas, it is essential to choose a proper basis ofcycles. In our text we use the one illustrated in Figure 7.36 EE E E E a b a E E b a E E a E E b E E E b b a a b Figure 6:
The system of cuts and basis of cycles used in our paper for N = 3 .The objects on Sheet 1 are drawn in solid lines, the objects on Sheet 2 are drawnin dashed lines. All cycles a j lie on Sheet 1. To calculate finite-gap solutions we need the periods of the basic holomor-phic differentials and some meromorphic differentials, the vector of Riemannconstants and the Abel transform of the Dirichlet spectrum. Let us calculatethem to the leading order. In our text we use the following agreements:37. The basic holomorphic differentials are normalized by: (cid:0) (cid:126)a k (cid:1) j = (cid:73) a j ω k = 2 πi. (76)In this normalization the real part of the Riemann matrix B = ( b jk ) b jk = (cid:0) (cid:126)b k (cid:1) j = (cid:73) b j ω k (77)is negative defined .2. In Sections 5.2, 5.3 we assume that the starting point for the Abeltransform is the branch point E = i + O ( (cid:15) ): (cid:126)A ( γ ) = (cid:126)A E ( γ ) = γ (cid:82) E ω ... γ (cid:82) E ω N = γ (cid:90) E (cid:126)ω, where (cid:126)ω = ω ... ω N , γ ∈ Γ . (78) The calculation of the diagonal elements to the leading order does not dependon N ; therefore we can use the results of the paper [48]: b jj = 2 log (cid:15) (cid:113) ˆ α j ˆ β j | φ j ) cos( ˆ φ j ) | + O ( (cid:15) ) , ≤ j ≤ N. (79)exp (cid:18) b jj (cid:19) = (cid:15) (cid:113) ˆ α j ˆ β j | φ j ) cos( ˆ φ j ) | + O ( (cid:15) ) , ≤ j ≤ N, where ˆ α j , ˆ β j are defined in (32). Up to O ( (cid:15) ) corrections, the off-diagonalelements do not depend on the perturbation, and coincide with the limitingvalues from [62]: b jk = 2 log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin (cid:16) ˆ φ j − ˆ φ k (cid:17) cos (cid:16) ˆ φ j + ˆ φ k (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O ( (cid:15) ) , j (cid:54) = k, j, k = 1 , . . . , N. (80)38 .3 The vector of Riemann constants The calculation of the Riemann constant vector is rather standard (see, forexample, [39]), but we present it for completeness. It is based on the period-icity properties of the Riemann theta-functions (65) θ ( (cid:126)A ( γ ) − (cid:126)C + (cid:126)a k ) = θ ( (cid:126)A ( γ ) − (cid:126)C ) (81) θ ( (cid:126)A ( γ ) − (cid:126)C + (cid:126)b k ) = θ ( (cid:126)A ( γ ) − (cid:126)C ) exp (cid:18) − b kk − A k ( γ ) + C k (cid:19) , (82)where (cid:126)C is a generic vector and C k are its components. Thereforelog (cid:2) θ ( (cid:126)A ( γ ) − C + (cid:126)a k ) (cid:3) = log (cid:2) θ ( (cid:126)A ( γ ) − C ) (cid:3) + 2 πiL k , L k ∈ Z , log (cid:2) θ ( (cid:126)A ( γ ) − C + (cid:126)b k ) (cid:3) == log (cid:2) θ ( (cid:126)A ( γ ) − C )] − b kk − A k ( γ ) + C k + 2 πiM k , M k ∈ Z , (cid:90) a k d log (cid:2) θ ( (cid:126)A ( γ ) − C ) (cid:3) = 2 πiL k , (cid:90) b k d log (cid:2) θ ( (cid:126)A ( γ ) − C ) (cid:3) = − b kk − A k (starting point of b k ) + C k + 2 πiM k ,d log (cid:2) θ ( (cid:126)A ( γ ) − C + (cid:126)a k ) (cid:3) = d log (cid:2) θ ( (cid:126)A ( γ ) − C ) (cid:3) ,d log (cid:2) θ ( (cid:126)A ( γ ) − C + (cid:126)b k ) (cid:3) = d log (cid:2) θ ( (cid:126)A ( γ ) − C ) (cid:3) − ω k . (83)Consider the 8 N gone obtained from Γ by cutting along the cycles. Todo such cutting it is important to have a non-intersecting system of cyclesstarting from the point E (see Figure 7).39 E E a E a E b E b a b E E a E E b b a a b a a b b b b a a b b a a E γ γ γ γ P =E a a a P b +1 a P b −1 b +2 b −2 P P P P P P P P P b −3 a b +3 P a a P P P a b −4 b +4 Figure 7: On the left: the system of cuts on Γ for N = 2. On the right above:the order of cycles near the point E . On the right below: the standard 8 N -gone for N = 2. The starting point is also connected with the zeroes γ ,. . . , γ N of the function θ ( (cid:126)A ( γ ) − (cid:126)C ) The order of the basic cycles along the8 N -gone agrees with the order of the cycles in the figure above.Let us remark that 40 A ( P k, ) = (cid:126) , (cid:126)A ( P k, ) = (cid:126)a k ,(cid:126)A ( P k, ) = (cid:126)a k + (cid:126)b k , (cid:126)A ( P k, ) = (cid:126)b k . (84)Denote by D = γ + . . . + γ N the divisor of zeroes of the function θ ( (cid:126)A ( γ ) − (cid:126)C ) (since (cid:126)C to be generic, this function is not identically equal to zero).Integrating the form ω j log (cid:2) θ ( (cid:126)A ( γ ) − (cid:126)C ) (cid:3) either along the paths connectingthe point E with the divisor points or along the 8 N -gone, we obtain: − πi (cid:126)A j ( D ) = (cid:90) ∂ Γ ω j log (cid:2) θ ( (cid:126)A ( γ ) − C ) (cid:3) = N (cid:88) k =1 (cid:34)(cid:90) a + k ω j ( γ ) log (cid:2) θ ( (cid:126)A ( γ ) − C ) (cid:3) + (cid:90) b + k ω j log (cid:2) θ ( (cid:126)A ( γ ) − C ) (cid:3) −− (cid:90) a − k ω j log (cid:2) θ ( (cid:126)A ( γ ) − C ) (cid:3) − (cid:90) b − k ω j log (cid:2) θ ( (cid:126)A ( γ ) − C ) (cid:3)(cid:35) == N (cid:88) k =1 (cid:34)(cid:90) a + k ω j log (cid:2) θ ( (cid:126)A ( γ ) − C ) (cid:3) − (cid:90) a + k ω j (cid:20) log (cid:2) θ ( (cid:126)A ( γ ) − C ) (cid:3) − b kk − A k ( γ ) + C k + 2 πiM k (cid:21) − (cid:90) b − k ω j d log (cid:2) θ ( (cid:126)A ( γ ) − C ) (cid:3) + (cid:90) b − k ω j (cid:2) log (cid:2) θ ( (cid:126)A ( γ ) − C ) (cid:3) + 2 πiL k (cid:3)(cid:35) = N (cid:88) k =1 (cid:34)(cid:90) a + k ω j (cid:20) b kk + A k ( γ ) − C k − πiM k (cid:21) + (cid:90) b − k ω j πiL k (cid:3)(cid:35) = 2 πi (cid:34) b jj − C j − πiM j + 2 πiL j + 12 πi N (cid:88) k =1 (cid:34)(cid:90) a + k ω j A k ( γ ) (cid:35)(cid:35) . Finally, modulo period, we obtain (cid:126)A j ( D ) = − b jj C j − πi N (cid:88) k =1 (cid:34)(cid:90) a + k ω j A k ( γ ) (cid:35) = C j − K j , j = b jj − πi + 12 πi N (cid:88) k =1 ,k (cid:54) = j (cid:90) a + k A k ( γ ) ω j . (85)In addition, if j (cid:54) = k , then (cid:90) a + k A j ( γ ) ω k = 2 πiA j (one of the branch points of a k );therefore, at last, K j = b jj − πi − N (cid:88) k =1 ,k (cid:54) = j (cid:26) A j ( E k ) , k ≤ N,A j ( E k − N ) − ) , k > N. (86)42rom a simple direct calculation it follows that (cid:126)A ( E ) = (cid:126) ,(cid:126)A ( E ) = 12 (cid:104) − (cid:126)b (cid:105) ,(cid:126)A ( E ) = 12 (cid:104) − (cid:126)b + (cid:126)a (cid:105) ,(cid:126)A ( E ) = 12 (cid:104) − (cid:126)b + (cid:126)a (cid:105) ,(cid:126)A ( E ) = 12 (cid:104) − (cid:126)b + (cid:126)a + (cid:126)a (cid:105) ,. . . (cid:126)A ( E k − ) = 12 (cid:34) k − (cid:88) j =1 (cid:126)a j − (cid:126)b k (cid:35) , k ≤ N,(cid:126)A ( E k ) = 12 (cid:34) k (cid:88) j =1 (cid:126)a j − (cid:126)b k (cid:35) , k ≤ N,(cid:126)A ( E ) = 12 (cid:34) N (cid:88) j =1 (cid:126)a j (cid:35) ,(cid:126)A ( E ) = 12 (cid:34) N (cid:88) j =1 (cid:126)a j − (cid:126)b N +1 (cid:35) ,(cid:126)A ( E ) = 12 (cid:34) N − (cid:88) j =1 (cid:126)a j − (cid:126)b N +1 (cid:35) ,. . . (cid:126)A ( E k − ) = 12 (cid:34) N − k +1 (cid:88) j =1 (cid:126)a j − (cid:126)b N + k (cid:35) , k ≤ N,(cid:126)A ( E k ) = 12 (cid:34) N − k (cid:88) j =1 (cid:126)a j − (cid:126)b N + k (cid:35) , k ≤ N. (87)We see that the a k and b k parts of (cid:126)K can be calculated separately: (cid:126)K =43 K a + (cid:126)K b . For (cid:126)K b we immediately obtain: (cid:126)K b = 12 N (cid:88) j =1 (cid:126)b j . Let us calculate (cid:126)K a . Modulo b k we have:12 = (cid:26) A j ( E j ) , j ≤ N,A j ( E j − N ) − ) , j > N ;therefore, modulo b k ,( K a ) j = − N (cid:88) k =1 (cid:26) A j ( E k ) , k ≤ N,A j ( E k − N ) − ) , k > N, and (cid:126)K a = πi , N odd; (cid:126)K a = πi , N even . Finally, we obtain: (cid:126)K = N (cid:88) k =1 (cid:2) (cid:126)A ( E k ) + (cid:126)A ( E k − ) (cid:3) . (88)Here we used the following argument. Each cycle a k consists of threeparts: a path P k, connecting E to the point E k − if k ≤ N , or to the44oint E k − N ) if k > N , a closed contour P k, going about either the interval[ E k − , E k ] or [ E k − N ) , E k − N ) − ] respectively, and the path P k, with theopposite orientation.Assume that k (cid:54) = j . Then the function A j ( γ ) has the same values when wego forward and back along P k, ; therefore this part of the integral vanishes: (cid:90) P k, A j ( γ ) ω k ( γ ) = (cid:90) P k, A j (BP) ω k ( γ ) + (cid:90) P k, [ A j ( γ ) − A j (BP)] ω k ( γ ) . The function A j ( γ ) − A j (BP) takes opposite values on the opposite sides ofthe cycle around the cut; therefore the second integral is equal to zero, and (cid:90) a k A j ( γ ) ω k ( γ ) = 2 πiA j (BP) , where BP denotes one of the branch points E k − , E k for k ≤ N , or one ofthe branch points E k − N ) , E k − N ) − for k > N . Let us also calculate (cid:90) a k A k ( γ ) ω k ( γ ) = 12 (cid:90) a k d (cid:0) A k ( γ ) (cid:1) = 12 (cid:2) A k (final point of the path) − A k (starting point of the path) (cid:3) = (2 πi ) D = γ + . . . + γ N of degree 2 N has the following structure: the point γ j is located near the point ˆ λ j . Then formula (63) admits the followingsimplification: (cid:126)A ( D ) + (cid:126)K = N (cid:88) j =1 γ j (cid:90) E j (cid:126)ω + N (cid:88) j =1 γ j + N (cid:90) E j − (cid:126)ω. (89) Agreement.
In the remaining part of our paper we use the following nota-tions: if a divisor point γ j is located close enough to the point ˆ λ j , then wecalculate its Abel transform starting from the corresponding branch point: (cid:126)A ( γ j ) = (cid:126)A E j ( γ j ) , (cid:126)A ( γ j + N ) = (cid:126)A E j − ( γ j + N ) , j = 1 , . . . , N ; (90)45herefore we can redefine the Abel transform of the divisor by: (cid:126)A ( D ) = N (cid:88) j =1 (cid:20) (cid:126)A E j ( γ j ) + (cid:126)A E j − ( γ j + N ) (cid:21) . (91)Formula (89) means that, if in the theta-functional formulas, the Abel trans-form of the divisor is calculated using the Agreement (91), then the vectorof Riemann constants (cid:126)K is identically zero: (cid:126)K = . (92)This simplification was not used in [48]. The formulas for finite-gap solutions include the following meromorphic dif-ferentials:1. A 3-rd kind meromorphic differential Ω with zero a -periods and firstorder poles at ∞ + , ∞ − :Ω = − dλλ + O (1) , at ∞ + , Ω = dλλ + O (1) , at ∞ − , (cid:73) a j Ω = 0 , j = 1 , . . . , N. From the Riemann bilinear relations (see [95]), it follows that: A j ( ∞ − ) − A j ( ∞ + ) = Z j = (cid:73) b j Ω . A sufficiently standard calculation (for N = 1 the details can be foundin [48])) implies: (cid:126)A ( ∞ − ) = − (cid:126)A ( ∞ + ) , (cid:0) (cid:126)A ( ∞ + ) (cid:1) j = πi i ˆ φ j + O ( (cid:15) ) . (93)46. The differential of the multivalued quasimomentum function p ( γ ) is the2-nd kind meromorphic differential dp (quasimimentum differentials)such that dp = − dλ + O (1) , at ∞ + ,dp = dλ + O (1) , at ∞ − , (cid:73) a j dp = 0 , j = 1 , . . . , N. It is convenient to use the Riemann bilinear relations: (cid:73) b j dp = − (cid:2) res ∞ + [ p ω j ] + res ∞ − [ p ω j ] (cid:3) . Again, a sufficiently standard calculation (for N = 1 see [48]) implies: (cid:73) b j dp = 2 cos( ˆ φ n ) + O ( (cid:15) ) . (94)Taking into account that (cid:16) (cid:126)U (cid:17) j = − i (cid:73) b j dp, (95)we immediately obatin (cid:16) (cid:126)U (cid:17) j = − i cos( ˆ φ j ) + O ( (cid:15) ) . (96) Remark 16
If the finite-gap approximation is constructed using someperiodicity preserving technique, for example the isoperiodic deforma-tion from [53], then, instead of (94), we have an exact relation: (cid:73) a j dp = 2 cos( ˆ φ n ) . (97) Otherwise, if we close the gaps corresponding to the stable modes “naively”,the spatial periodicity of the leading order solution breaks at O ( (cid:15) ) .
47. A 2-nd kind meromorphic differential dq such that dq = − d ( λ ) + O (1) , at ∞ + ,dq = d ( λ ) + O (1) , at ∞ − , (cid:73) a j dq = 0 , j = 1 , . . . , N. Again, using the Riemann bilinear relations: (cid:73) b j dq = − (cid:2) res ∞ + [ q ω j ] + res ∞ − [ q ω j ] (cid:3) , we obtain (for N = 1 see [50]) (cid:73) b j dq = i sin(2 ˆ φ j ) + O ( (cid:15) ) . (98)Taking into account that (cid:16) (cid:126)U (cid:17) j = − i (cid:73) b j dq, (99)we immediately obatin (cid:16) (cid:126)U (cid:17) j = 2 sin(2 ˆ φ j ) + O ( (cid:15) ) . (100) In this Section we calculate the Abel transform of divisor points assumingnormalization (90) up to O ( (cid:15) ) correction.First of all, under this assumption, A k (ˆ γ j ) = O ( (cid:15) ) , k (cid:54) = j. and we will neglect it.Let us calculate A j (ˆ γ j ). 48ear the point ˆ λ j we have: p ( γ ) = p ( E j ) + ˆ λ j ˆ µ j ν ( γ ) + O ( (cid:15) ) , j ≤ N,p ( E j − N − ) + ˆ λ j ˆ µ j ν ( γ ) + O ( (cid:15) ) , j > N. where ν = (cid:26) ( λ − E j − )( λ − E j ) , j ≤ N, ( λ − E j − N − )( λ − E j − N ) , j > N. We also have: ω j = dλν + O ( (cid:15) ) = d log (cid:104) λ ( γ ) − ˆ λ j + ν ( γ ) (cid:105) + O ( (cid:15) )and A j (ˆ γ j ) = log (cid:34) λ (ˆ γ j ) − ˆ λ j + ν (ˆ γ j ) E s − ˆ λ j (cid:35) , + O ( (cid:15) )where E s = (cid:26) E j , j ≤ N,E j − N − , j > N. We have, up to O ( (cid:15) ) corrections: E s − ˆ λ j = (cid:15) λ j (cid:112) α j β j , j ≤ N, − (cid:15) λ j (cid:113) ˆ α j ˆ β j , j > N. Here we asssume:Re (cid:112) α j β j ≥ , (cid:113) ˆ α j + N ˆ β j + N = (cid:112) α j β j . Using (73), (74), we obtain: λ (ˆ γ j ) − ˆ λ j + ν (ˆ γ j ) = (cid:40) e i ˆ φ j (cid:15) λ j ˆ α j + O ( (cid:15) ) , j ≤ N, − e i ˆ φ j (cid:15) λ j ˆ α j + O ( (cid:15) ) , j > N, and, finally, A j (ˆ γ j ) = log ˆ α j (cid:113) ˆ α j ˆ β j + i ˆ φ j + O ( (cid:15) ) . Theta-functional solutions
The finite-gap leading order solution of the Cauchy problem of the anomalouswaves is provided by formula (63), with C = θ ( (cid:126)A ( ∞ + ) − (cid:126)A ( D ) | B ) θ ( (cid:126)A ( ∞ − ) − (cid:126)A ( D ) | B ) u (0 , O ( (cid:15) )) , U = O ( (cid:15) ) , V = 2 i + O ( (cid:15) ) . (101)Taking into account Agreement (91), and using (92), (93), (96), (100), weobtain the following approximation for the arguments of the theta-function: (cid:126)A ( ∞ + ) − (cid:126)U x − (cid:126)U t − (cid:126)A ( D ) − (cid:126)K = (cid:126)z − ( x, t ) + O ( (cid:15) ) ,(cid:126)A ( ∞ − ) − (cid:126)U x − (cid:126)U t − (cid:126)A ( D ) − (cid:126)K = (cid:126)z + ( x, t ) + O ( (cid:15) ) , (cid:0) (cid:126)z − ( x, t ) (cid:1) j = iπ − log ˆ α j (cid:113) ˆ α j ˆ β j + 2 i cos( ˆ φ j ) x − φ j ) t (102) (cid:0) (cid:126)z + ( x, t ) (cid:1) j = (cid:0) (cid:126)z − ( x, t ) (cid:1) j − πi − i ˆ φ j , (103)and, finally, we obtain the leading order solution u ( x, t ) = exp(2 it ) · θ ( (cid:126)z + ( x, t ) | B ) θ ( (cid:126)z − ( x, t ) | B ) · (1 + O ( (cid:15) )) . (104)in terms of the genus 2 N θ -functions.
Formula (104) provides the solution up to O ( (cid:15) ) corrections. Therefore it isenough to sum the exponents in the theta-function over the elementary hy-percube in R N containing the trajectory point − (cid:126)w ( t ), where (cid:126)w ( t ) is definedin (37), and we obtain: θ ( (cid:126)z ± ( x, t ) | B ) = ˜ θ ( (cid:126)z ± ( x, t ) | B )(1 + O ( (cid:15) )) , ˜ θ ( (cid:126)z ± ( x, t ) | B ) = (cid:88) n min j ( t ) ≤ n j ≤ n max j ( t ) j = 1 , , . . . , N exp (cid:34) N (cid:88) l =1 2 N (cid:88) s =1 b ls n l n s + N (cid:88) l =1 n l ( (cid:126)z ± ( x, t )) l (cid:35) , (105)50 min j ( t ) = −(cid:98) w j ( t ) + 1 (cid:99) , n max j ( t ) = −(cid:98) w j ( t ) (cid:99) . Here the floor function (cid:98) x (cid:99) denotes the largest integer less of equal to x .If we are interested in constructing the solution up to order | (cid:15) | p correc-tions, 0 < p <
1, for generic t , only a subset of vertices of that hypercubecontributes. Therefore formula (105) admits a further simplification.To estimate the summands in the θ -functions expansions, we use thefollowing identity:Re (cid:32) N (cid:88) l =1 2 N (cid:88) s =1 b ls n l n s + 2 N (cid:88) l =1 n l ( z ± ( x, t )) l (cid:33) = N (cid:88) l =1 2 N (cid:88) s =1 Re( b ls ) n l n s +2 N (cid:88) l =1 n l Re( z ± ( x, t )) l == N (cid:88) l =1 2 N (cid:88) s =1 Re( b ls )( n l + w l )( n s + w s ) − N (cid:88) l =1 2 N (cid:88) s =1 Re( b ls ) w l w s . (106)Consider the metric on R N with the metric tensor g kl = Re b kl . Denote thedistance between the point − ˜ w ( t ) and the closest vertex of this hypercubeby d . Relation (106) means that the real parts of the arguments of theexponent in the θ -series are equal, up to a common constant (the secondterm in the right hand side of (106)), to the distance between the point − ˜ w and the corresponding lattice point. Therefore it is sufficient to keep in (105)only the vertices (cid:126)n such that the distance between − ˜ w ( t ) and (cid:126)n is smallerthan a critical distance: d cr = d + pπ | log( (cid:15) ) | , and in formula (105) we sum only over the corresponding 2 N ( t ) dimensionalsub-cube of the full hypercube, N ( t ) ≤ N , where 2 N ( t ) is the number of j ’ssuch that ˜ n min j ( t ) < ˜ n max j ( t ), with:˜ n min j = −(cid:98) w j ( t ) + 1 / p/ (cid:99) , ˜ n max j ( t ) = −(cid:100) w j ( t ) − / − p/ (cid:101) . Here the ceiling function (cid:100) x (cid:101) denotes the smallest integer greater or equalto x . It is easy to verify that this number is always even. Restricting thesummation to this subset of vertices means that we approximate the NLSsolution by the exact N ( t )-soliton solution of Akhmediev type (see formula(48)), whose parameters are written down through the Cauchy data in termsof elementary functions. 51or numerical simulations, the representation (105) is not very convenientbecause it involves ratios of exponentials with big arguments. To avoid thisproblem, one can use the periodicity properties of the theta-functions (65)and shift the arguments to the basic elementary cell:(˜ z ± ) j ( x, t ) = ( (cid:126)z ± ( x, t )) j − (cid:88) k b jk (cid:98) w j ( t ) (cid:99) . Introducing also: ˜ w j ( t ) = w j ( t ) − (cid:98) w j ( t ) (cid:99) , then equation (104) becomes u ( x, t ) = exp (2 it + 2 i Φ) θ (˜ z + ( x, t ) | B ) θ (˜ z − ( x, t ) | B ) × (1 + O ( (cid:15) )) , (107)where Φ = N (cid:88) j =1 ( π φ j ) (cid:98) w j ( t ) (cid:99) . If the argument z belongs to the basic elementary cell, to obtain the O ( (cid:15) )approximation, we sum over the exponentials of the fundamental hypercube:˜ θ ( z | B ) = (cid:88) n j ∈ {− , } j = 1 , , . . . , N exp (cid:34) πi (cid:32) N (cid:88) l =1 2 N (cid:88) s =1 b ls n l n s + 2 N (cid:88) l =1 n l z l (cid:33)(cid:35) . (108)Equivalently, shifting the hypercube by 1 / n j = 2 n j + 1),one can write: u ( x, t ) = exp (2 it + 2 i Φ) ˆ θ (ˆ z + ( x, t ) | B )ˆ θ (ˆ z − ( x, t ) | B ) (1 + O ( (cid:15) )) , ˆ θ (ˆ z | B ) = (cid:88) ˆ n j ∈ {− , } j = 1 , , . . . , N exp N (cid:88) l, s = 1 l (cid:54) = s log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin (cid:16) ˆ φ l − ˆ φ s (cid:17) cos (cid:16) ˆ φ l + ˆ φ s (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ n l ˆ n s N (cid:88) l =1 ˆ n l ˆ z l − N (cid:88) l =1 ˆ z j , (109)52here ˆ z j = ˜ z j − (cid:88) k b jk . If the number of unstable modes is not too large (see Remark 11), theoff-diagonal terms in B could be omitted in the above rule for selecting thesubset of hypercube vertices providing the main contribution (but, of course,the off-diagonal terms should be kept in the arguments of the exponents in(109)).Then we obtain the following rule:1. If ˜ w j < − p we keep only the terms with n j = 1.2. If − p ≤ ˜ w j ≤ p we keep the terms with n j = 1 and with n j = − w j > p we keep only the terms with n j = − t , we have a summation over the vertices of ahypercube of dimension 2 N ( t ), N ( t ) ≤ N . In our approximation the off-diagonal terms of matrix B do not depend on (cid:15) ; therefore, in each timeinterval I , the approximation function does not depend on (cid:15) , and in eachtime interval we approximate the solutions by an exact NLS solution, the N ( I )-soliton solution of Akhmediev type (47)-(50) (see Figures 9).Suppose that, in some time interval I , only the j th unstable mode is visible(for example, in Figure 8, only the first mode k is visible, and appears asan isolated peak in the center of the picture). Then, in the interval I , thepair of variables ˜ w j , ˜ w j + N is close to 1 / w k are either close to0 or to 1. Then we have the following approximate formulas for the position( X max , T max) of the isolated j th mode: T max = T (1) j + M j ∆ T j + (cid:88) ≤ k ≤ Nk (cid:54) = j M k ∆ T ( j, k ) , (110) X max = X (1) j + M j ∆ X j . (111) T (1) j , X (1) j would be the coordinates of the first appearance of the j th mode,and ∆ T j and ∆ X j would be the recurrence time and the x shift of j th mode,53f it did not interact with the other modes, and therefore are defined in (16). M k indicates how many times the mode k was visible before the time T max : M k = (cid:26) nearest integer to w k , k (cid:54) = j, (cid:98) w j (cid:99) k = j, (112)and ∆ T ( j, k ) = 1 σ j log (cid:12)(cid:12)(cid:12)(cid:12) sin( φ j + φ k )sin( φ j − φ k ) (cid:12)(cid:12)(cid:12)(cid:12) (113)is the time delay in the appearance of the j th mode due to its pairwiseinteraction with the k th , j (cid:54) = k mode. We see that that the presence of theother modes delays the appearance of the j -th mode (see (110)), but does notaffect the x -shift (see (111)). We also see that the interaction of the unstablemodes is pairwise. 54igure 8: The graph of the finite-gap approximation of | u ( x, t ) | , with L = 14(4 unstable modes), 0 ≤ t ≤ c = 0 . c − = 0 . . i , c = 0 . c − = − .
03 + 0 . i , c = 0 . c − = 0 . . i , c = 0 . c − = − . . i , (cid:15) = 10 − . The graph below shows how the number N ( t ) depends on t . Thegraphs obtained by numerical integration, by applying the full-hypercubefinite-gap approximation (108), and the graph obtained using the relevantvertices only are identical pixel-to-pixel; therefore we show only one of them.We see that, after reaching the first nonlinear stage of MI, we do not returnto the pure background state in the time interval studied. The appearancetime of single peak in the middle of the graph is T max = 18 . T max = 18 . T max = 18 . L = 20 (6 unstable modes), 0 ≤ t ≤ c = 0 . c − = 0 . . i , c = 0 . c − = − .
03 + 0 . i , c = 0 . c − =0 . . i , c = 0 . c − = − . . i , c = 0 . c − = 0 . . i , c =0 . c − = − . . i , (cid:15) = 10 − , p = 1 /
2. The figure above shows thegraph of | u ( x, t ) | obtained using the analytic formula (107), coinciding pixel-to-pixel with numerical simulation. The middle graph shows the absolutevalue of the difference between the numerical solution and the full-hypercubefinite-gap approximation, multiplied by 10 . The difference at the left-handside of the picture, of order 10 − , is very likely to be a numerical artifact.The graph below shows the absolute value of the difference between thefull-hypercube finite-gap approximation and the approximation involving therelevant vertices only, multiplied by 10 ; its magnitude is O (10 − ), a littlehigher than (cid:15) / but, taking into account that the full hypercube contains4 = 4096 points, the agreement is sufficiently good (see Remark 11).56 Acknowledgments
The work P. G. Grinevich was supported by the Russian Science Foundationgrant 18-11-00316. P. M. Santini was partially supported by the UniversityLa Sapienza, grant 2017.We acknowledge useful discussions with F. Calogero, C. Conti, E. DelRe, A.Degasperis, D. Pierangeli, M. Sommacal, and V. Zakharov.
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