Numerical spectral synthesis of breather gas for the focusing nonlinear Schrödinger equation
Giacomo Roberti, Gennady El, Alexander Tovbis, François Copie, Pierre Suret, Stéphane Randoux
NNumerical spectral synthesis of breather gas for the focusing nonlinear Schr¨odingerequation
Giacomo Roberti, Gennady El, Alexander Tovbis, Fran¸cois Copie, Pierre Suret, and St´ephane Randoux ∗ Department of Mathematics, Physics and Electrical Engineering,Northumbria University, Newcastle upon Tyne, NE1 8ST, United Kingdom Department of Mathematics, University of Central Florida, Orlando, USA Univ. Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers Atomes et Mol´ecules, F-59 000 Lille, France
We numerically realize breather gas for the focusing nonlinear Schr¨odinger equation. This is doneby building a random ensemble of N ∼
50 breathers via the Darboux transform recursive schemein high precision arithmetics. Three types of breather gases are synthesized according to the threeprototypical spectral configurations corresponding the Akhmediev, Kuznetsov-Ma and Peregrinebreathers as elementary quasi-particles of the respective gases. The interaction properties of theconstructed breather gases are investigated by propagating through them a ‘trial’ generic breather(Tajiri-Watanabe) and comparing the mean propagation velocity with the predictions of the recentlydeveloped spectral kinetic theory (El and Tovbis, PRE 2020).
I. INTRODUCTION
The study of nonlinear random waves in physical sys-tems well described at leading order by the so-called in-tegrable equations, such the Korteweg-de Vries (KdV)or nonlinear Schr¨odinger (NLS) equations has recentlybecome the topic of intense research in several areas ofnonlinear physics, notably in oceanography and nonlin-ear optics. This interest is motivated by the complex-ity of many natural or experimentally observed nonlin-ear wave phenomena often requiring a statistical descrip-tion even though the underlying physical model is, inprinciple, amenable to the well-established mathematicaltechniques of integrable systems theory such as inversescattering transform (IST) or finite-gap theory (FGT)[1]. An intriguing interplay between integrability andrandomness in such systems is nowadays associated withthe concept of integrable turbulence introduced by V. Za-kharov in [2]. The integrable turbulence framework isparticularly pertinent to the description of modulation-ally unstable systems whose solutions, under the effectof random noise, can exhibit highly complex spatiotem-poral dynamics that are adequately described in termsof turbulence theory concepts, such as the distributionfunctions, ensemble averages, correlations etc.Solitons and breathers are the elementary “quasiparti-cles” of nonlinear wave fields in integrable systems whichcan form ordered coherent structures such as modulatedsoliton trains and dispersive shock waves [3, 4], “su-perregular breathers” [5, 6] or “breather molecules” [7].Furthermore, solitons and breathers can form irregular structures or statistical ensembles that can be viewed assoliton and breather gases. The nonlinear wavefield insuch integrable gases represents a particular case of inte-grable turbulence [2, 8–13]. The observations of solitonand breather gases in the ocean have been reported in ∗ [email protected] [14–16]. Recent laboratory experiments on the genera-tion of shallow-water and deep water soliton gases werereported in [17] and [18] respectively. It has also beendemonstrated that the soliton gas dynamics in the focus-ing NLS equation provides a remarkably good descrip-tion of the statistical properties of the nonlinear stage ofspontaneous modulational instability [19].Analytical description of soliton gases was initiated byZakharov in ref. [20], where a spectral kinetic equationfor KdV solitons was derived using an IST based phe-nomenological procedure of computing an effective ad-justment to a soliton’s velocity in a rarefied gas due toits collisions with other solitons, accompanied by appro-priate phase-shifts. Zakharov’s kinetic equation for KdVsoliton has been generalized to the case of a dense gas inref [21] using the spectral finite-gap theory. Within thistheory, a uniform (equilibrium) soliton gas is modelledby a special infinite-phase, thermodynamic type limit offinite-gap KdV solutions. The kinetic description of thenon-equilibrium soliton gas is then enabled by consid-ering the same thermodynamic limit for the associatedmodulation (Whitham) equations. The resulting kineticequation describes the evolution of the density of states(DOS) defined as the density function in the spectral(IST) phase plane of soliton gas. The spectral construc-tion of the KdV soliton gas in ref. [21] has been gener-alized to the soliton gas of the focusing NLS equation in[22, 23]. The latter work [23] provides also the spectralkinetic description of a breather gas (BG), which is themain subject of the present work.An isolated generic breather can be broadly viewed asa soliton on the plane wave (or finite) background. The1D-NLSE equation supports a large family of breathersolutions that have attracted a particular interest due totheir explicit analytic nature and the potential for mod-eling the rogue wave events in the ocean and in non-linear optical fibers [24–28]. Three types of breathers,namely the Akhmediev breather (AB), the Kuznetsov-Ma (KM) breather and the Peregrine soliton (PS) havearoused significant research interest, see [29–34] and ref- a r X i v : . [ n li n . PS ] J a n erences therein. AB, KM breather and PS represent spe-cial cases of a generic breather called the Tajiri-Watanabe(TW) breather [35]. A simplest example of breather gascan be viewed as an infinite random ensemble of the TWbreathers [23]. By manipulating the spectral parametersthe TW breather gas can be reduced to the AM, KMand PS gases as well as to the fundamental soliton gas.The latter is achieved by vanishing the plane wave back-ground of the TW breather gas [23].The present paper has two goals: (i) numerical real-ization of a breather gas; (ii) verification of the spectraltheory of breather gas developed in [23].Numerical realization of a breather gas as a large en-semble of TW breathers with prescribed parameters rep-resents a challenging problem. Numerical methods forthe construction of breather solutions of the 1D-NLSEsuffer from accuracy problems that prevent the numeri-cal synthesis of breathers of order N (cid:38) N ∼
50 breathers can be build viathe Darboux transform recursive scheme in high preci-sion arithmetics. To our knowledge, this represents animprovement of an order of magnitude compared to theresults reported in previous numerical works. In additionwe show that the construction method can be used to pro-vide evidence of the space-time evolution of the generatedbreather gases. This feature cannot be achieved by us-ing direct numerical simulations of the 1D-NLSE due tothe inevitable presence of modulational instability thatquickly desintegrates the plane wave background.The paper is organized as follows. In Section II wepresent the algorithm of the spectral synthesis of breathergas using the Darboux transform. This algorithm is thenrealized numerically using the high precision arithmetics.In Section III we numerically study the interactions inbreather gases and compare the results of the numer-ical simulations with the theoretical predictions of thebreather gas kinetic theory of Ref [23]. Specifically, weconsider the propagation of the ‘trial’ breather througha homogeneous breather gas for three prototypical con-figurations: Akhmediev, Kuznetsov-Ma and Peregrinegases. The study of interaction in the gas of Akhmedievbreathers has revealed some special features that have re-quired further development of the theory of Ref [23]. TheAppendix provides the identification of the interactionkernel in the breather gas with the position shift formulain two-breather collisions, obtained in earlier works.
II. NONLINEAR SPECTRAL SYNTHESIS OFBREATHER GASESA. Soliton and breather ensembles in the1D-NLSE: an overview
We consider the integrable one-dimensional focusingNLS equation (1D-NLSE) in the following form: iψ t + ψ xx + 2 | ψ | ψ = 0 , (1)where ψ ( x, t ) represents the complex envelope of the wavefield that evolves in space x and time t .In the inverse scattering transform (IST) method, the1D-NLSE (1) is represented as a compatibility conditionof two linear equations [1, 39],Φ x = (cid:18) − iλ ψ − ψ ∗ iλ (cid:19) Φ , (2)Φ t = (cid:18) − iλ + i | ψ | iψ x + 2 λψiψ ∗ x − λψ ∗ iλ − i | ψ | (cid:19) Φ , (3)where λ is a (time-independent) complex spectral param-eter and Φ( x, t, λ ) = ( r ( x, t, λ ) , s ( x, t, λ )) T is a columnvector. The spatial linear operator (2) and the temporallinear operator (3) form the Lax pair of Eq. (1). For agiven potential ψ ( x, t ) the problem of finding the scatter-ing data σ [ ψ ] (also sometimes called the IST spectrum)and the corresponding scattering solution Φ specified bythe spatial equation (2) is called the Zakharov-Shabat(ZS) scattering problem [40]. The ZS scattering problemis formally analogous to calculating the Fourier coeffi-cients in Fourier theory of linear systems, hence the term‘Nonlinear Fourier Transform’ is often used in the contextof telecommunications systems research, particularly inthe context of periodic boundary conditions [41–43].For spatially localized potentials ψ such that ψ ( x, t ) → | x | → ∞ , the complex eigenvalues λ are gener-ally presented by a finite number of discrete points with (cid:61) ( λ ) (cid:54) = 0 (discrete spectrum) and the real line λ ∈ R (continuous spectrum). The scattering data σ ( ψ ) consistof a set of N discrete eigenvalues λ n ( n = 1 , ..., N ) , a setof N norming constants C n for each λ n and the so-calledreflection coefficient ρ ( ξ ), σ ( ψ ) = { ρ ( ξ ); λ n , C n } (4)where ξ ∈ R denotes the continuous spectrum compo-nent. In this setting where the wavefield ψ lives on a zerobackground (ZBG), the discrete part of the IST spectrumis related to the soliton content of the wavefield whereasthe continuous part of the IST spectrum is related to thenonlinear dispersive radiation [40].A special class of (reflectionless) solutions of Eq. (1),the N -soliton solutions (N-SS’s), exhibits only a discretespectrum ( ρ ( ξ ) = 0) consisting of N complex-valuedeigenvalues λ n , n = 1 , ..., N and N associated complex-valued norming constants. The IST formalism has beenextensively applied to examine the processes of inter-action, collision and synchronisation in N-SS’s, see e.g.ref. [40, 44]. The numerical synthesis of N-SS’s can beachieved in standard computer simulations (double preci-sion, 16-digits) up to N ∼
10 [38]. On the other hand thenumerical synthesis of N-SS’s with N large represents achallenging problem that has been resolved only recently[38]. Combining the so-called dressing method and nu-merical calculations made using high numerical precision(a 100-digits precision is typically necessary for the syn-thesis of N-SS’s with N ∼ σ [ ψ ] in the IST with NZBC consist of a set of N discrete complex-valued eigenvalues λ n , a set of N associ-ated norming constants C n and the reflection coefficient ρ ( λ ). In IST with NZBC, the continuous spectrum doesnot live on the real axis R but on R ∪ [ − iq , iq ] where q > ρ ( λ ) = 0) named breathers or sometimes solitonson finite background. The generic “elementary” breatherparametrized by one single complex-valued eigenvalue( N = 1) in the framework of IST with NZBC is theso-called Tajiri-Watanabe breather [35]. This elemen-tary solution reduces under certain limits to the solu-tions found over the years by Kuznetsov and Ma [29, 47],Peregrine [30], and Akhmediev [31]. Using the dress-ing method, Zakharov and Gelash constructed a classof two-soliton solutions on finite background, termed su-perregular breathers and corresponding to small initialperturbations of a constant background [48]. This wasgeneralized to several pairs of breathers in ref. [5, 49].Note that most of these breather solutions of Eq. (1)have been experimentally realized in hydrodynamics andin optics [6, 7, 32, 33, 50–55]. B. Darboux transform-based synthesis of breathergases
The recent interest in studying the breather solutionsof various kind has been fuelled by the rogue wave re-search, see e.g. [56] and references therein. The pro-totype rogue-wave solutions represent coherent struc-tures of large amplitude, strongly localized in bothspace and time, on an otherwise quiescent background[24, 26, 37, 57–63]. In this context the Darboux transformhas been extensively used as a reliable method to gener-ate higher-order breather solutions of Eq. (1), i.e. reflec-tionless solutions of the focusing 1D-NLSE with NZBC[36, 64–67]. Note that the Darboux transform is now alsoused in the context of nonlinear eigenvalue communica-tion to build ordered soliton ensembles used to carry outthe transmission of information in fiber optics communi-cation links [42, 43, 68].The Darboux method is a recursive transformationscheme where a “seeding solution” of the focusing 1D-NLSE is used as a building block for the constructionof a higher-order solution through the addition of onediscrete eigenvalue. Here we give a brief review of theDarboux transform method used for the generation ofhigher-order breathers. We largely follow the expositiongiven in ref. [37, 69] but other important references wherethis method is described and used are ref. [36, 64–67].In the IST for the 1D-NLSE with NZBC, the seedingsolution commonly used at the first step of the recur-sive process of constructing a higher-order breather so-lution is the plane wave solution of Eq. (1) with unitamplitude, i.e. ψ ( x, t ) = e i t . The first-order breather(Tajiri-Watanabe) ψ ( x, t ) parametrized by the complexeigenvalue λ is obtained by ψ ( x, t ) = ψ ( x, t ) + 2( λ ∗ − λ ) s , r ∗ | r , | + | s , | . (5)The functions r , ( x, t ) and s , ( x, t ) in Eq. (5) are ob-tained by setting j = 1 in the following expressions r ,j ( x, t ) = 2 ie − it sin( A j ) ,s ,j ( x, t ) = 2 e it cos( B j ) , (6)where A j and B j are given by A j = 12 (cid:16) arccos (cid:16) κ j (cid:17) + ( x − x j ) κ j − π (cid:17) +( t − t j ) κ j λ j ,B j = 12 (cid:16) − arccos (cid:16) κ j (cid:17) + ( x − x j ) κ j − π (cid:17) +( t − t j ) κ j λ j , (7)with κ j = 2 (cid:113) λ j . The parameters ( x j , t j ) are con-nected with the complex norming constants C j in theIST with NZBC [36]. The first-order breather ψ ( x, t )is parametrized by the complex eignevalue λ and bythe two real parameters x and t . Once the first-order Figure 1. Numerical synthesis of a generic BG (left column (a), (e), (i)) and of three single-component BGs: a KM-BG (secondcolumn (b), (f), (j)), a AB-BG( third columnn (c), (g), (k)) and a PS-BG (fourth column (d), (h), (i)). The four BGs areparametrized by N = 50 complex eigenvalues λ n , see bottom row. The first row (a)–(d) represents the space-time evolutionof the BGs, with the second row (e)–(h) being an enlarged view of some restricted region of the ( x − t ) plane. The third row(i)–(l) represents the spectral portraits of each BG with the vertical line between 0 and + i being the branchcut associated withthe plane wave background. Each point in the upper complex plane in (i), (j), (k), (l) represents a discrete eigenvalue in theIST problem with NZBC. The eigenvalues parametrizing the single-component BGs are densely placed in a small square regionwhich is centered around a point λ of the imaginary vertical axis and which is strongly enlarged in the insets shown in (j),(k), (l). The x j are uniformly distributed in the range [ − ,
1] for the generic gas (a) and for the Peregrine gas (d) while theyare uniformly distributed in the range [ − ,
32] for the KM gas (b) and the AB gas (c). breather ψ is constructed using Eqs. (5), (6), (7),breather solutions of order n ≥ ψ n ( x, t ) = ψ n − ( x, t ) + 2( λ ∗ n − λ n ) s n, r ∗ n, | r n, | + | s n, | (8)with r n,p = [( λ ∗ n − − λ n − ) s ∗ n − , r n − , s n − ,p +1 +( λ p + n − − λ n − ) | r n − , | r n − ,p +1 +( λ p + n − − λ ∗ n − ) | s n − , | r n − ,p +1 ] / ( | r n − , | + | s n − , | ) , (9) s n,p = [( λ ∗ n − − λ n − ) s n − , r ∗ n − , r n − ,p +1 +( λ p + n − − λ n − ) | s n − , | s n − ,p +1 +( λ p + n − − λ ∗ n − ) | r n − , | s n − ,p +1 ] / ( | r n − , | + | s n − , | ) . (10)Despite the efficiency of the Darboux method for theconstruction of high-order breather solutions of Eq. (1), its practical implementation in numerics suffers from thesame type of issues as those previously mentioned forthe numerical construction of N-SS’s. As noted in ref.[36, 37], problems of numerical accuracy may preventthe numerical synthesis of breathers of order N (cid:38) N ∼
50. As will be shown in detailin Sec. III, this provides a numerical tool that enablesone to verify the results of the spectral theory of breathergases recently developed in ref. [23].Fig. 1(a) shows the space-time evolution of a genericBG, i.e. a breather solution of Eq. (1) of order N = 50with random spectral charateristics. The amplitude ofthe plane wave background is unity ( q = | ψ | = 1)and the 50 complex-valued eigenvalues λ j ( j = 1 − t j are fixed to zero( t j = 0 ∀ j ) and the randomness of the gas is achievedby uniformly distributing the x j in some interval centeredaround x = 0. Note that the vertical line between 0 and+ i in Fig. 1(i) represents the so-called branchcut associ-ated with the plane wave background in the IST formal-ism of the 1D-NLSE with NZBG, see e.g. [23, 34, 45, 46].Fig. 1(a) reveals that the space-time dynamics of thegeneric BG synthesized in numerical simulations is highlycomplicated. In particular, breathers cannot be indi-vidualized due to their strong overlap and interaction.Note also that the maximum amplitude reached locallyin space and time by the incoherent breather ensembleof Fig. 1(a) does not exceed ∼ .
5, which demonstratesthat the multiple breathers are far from a synchroniza-tion state that would eventually produce isolated roguewaves of large amplitude [70, 71].We emphasize that BGs shown in the space-time plotsof Fig. 1 are not obtained from a numerical simulationof Eq. (1). Taking a BG generated at a given time t using the Darboux method and using this wavefield asinitial condition in a numerical simulation of Eq. (1), weobserve that modulation instability quickly desintegratesthe plane wave background by amplifying the numeri-cal noise inherent to any pseudo-spectral (split-step like)method commonly used for the numerical integration ofthe 1D-NLSE. On the other hand space-time plots re-ported in Fig. 1 are obtained from a pure spectral (IST)construction based on the Darboux recursive methodwhich has been implemented in computer simulationsmade with high numerical precision. Starting from anensemble of N complex eigenvalues λ j and N coordinates( x j , t j ), the BG is synthesized at time t using the Dar-boux machinary (Eqs. (5)-(10)). A 100 digits precisionis typically necessary to synthesize a BG parametrizedby an ensemble of N ∼
50 eigenvalues. The space-timeplots shown in Fig. 1 are obtained by reiterating thesame synthesis at different values of time t .The central concept in the theory of SGs and BGs isthe density of states (DOS) [72] which represents the dis-tribution function u ( λ, x, t ) in the spectral phase space.In the context of the 1D-NLSE (1) the DOS u ( λ, x, t ),where λ = β + iγ , is defined such that udβdγdx is thenumber of breather states with complex spectral param-eter λ ∈ [ β, β + dβ ] × [ γ, γ + dγ ] contained in a portionof BG within a spatial interval [ x, x + dx ] at time t .One-component BGs have been defined in ref. [23] asbeing characterized by a DOS in the form of the Dirac δ distribution, i.e. u ( λ ) = w δ ( λ − λ ) where w > δ distribution which is cen-tered around one specfic point λ in the complex spectralplane. Fig. 1(b-d)(f-h) display the space-time evolutionstogether with the spectral portraits (Fig. 1(j-l)) typifyingsome one-component BGs of particular interest.For the Kuznetsov-Ma BG (KM-BG), the spectral por-tait consists of the branchcut (associated with the planewave background of unity amplitude) and a dense set of N = 50 spectral points randomly placed in a small square region of width δ = 10 − centered around λ = 1 . i , asshown in Fig. 1(j). Fig. 1(b) shows that the KM-BG isa dense ensemble of individual KM breathers having alla zero velocity in the ( x, t )-plane. Contrary to Fig. 1(a)each KM breather inside the BG can be individualizedand it follows the same periodic time evolution where thetime period is fully determined by (cid:61) ( λ ). The random-ness in the one-component KM-BG can be seen from therandom distance between individual KM breathers andtheir random initial phase, see Fig. 1(f).The Akhmediev BG (AB-BG) is characterized by thesame distribution of the spectrum λ as the KM-BG ex-cept that the point λ around which the multiple discreteeigenvalues are accumulated is now placed inside thebranchcut associated with the plane wave background,see Fig. 1(k) where λ = +0 . i . As a result, the AB-BGis more naturally characterized by the spectral flux den-sity , the temporal counterpart of the DOS. As shown inFig. 1(c), the AB-BG consists of a random series of indi-vidual ABs having identical spatial period, which is fullydetermined by (cid:61) ( λ ). Similarly to the KM-BG, the ran-domness in the one-component AB-BG can be seen fromthe random time separation between individual Akhme-diev breathers and their random relative phases, see Fig.1(g).It must be mentioned that the density (spatial ortemporal) of the AB or KM breather gases cannot bemade arbitrary large: there is a configuration termed“breather condensate” [23] corresponding to a criticallydense breather gas, similar to a soliton condensate nu-merically realized in [19].It is well known that the Peregrine breather can beobtained as the spatial and temporal infinite periodlimits of Akhmediev and Kuznetsov-Ma breathers re-spectively [66, 69]. In the spectral (IST) domain, thePeregrine breather is obtained by placing the eigenvalueparametrizing a first-order breather solution of Eq. (1)exactly at the endpoint + i of the branchcut associatedwith the plane wave background of unit amplitude [34].Following the same approach, the one-component Pere-grine BG (P-BG) is obtained by accumulating a largenumber of discrete eigenvalues in a small area surround-ing the endpoint of the branchcut, see Fig. 1(l). Asshown in Fig. 1(d) and in Fig. 1(h), the P-BG repre-sents a collection of individual and identical Peregrinebreathers that are randomly positioned in space andtime. III. INTERACTIONS IN BREATHER GASES:COMPARISON BETWEEN NUMERICALEXPERIMENTS AND SPECTRAL THEORY
The analytical theory of BGs has been introduced anddeveloped in ref. [23]. It has been shown that spa-tially non-homogeneous BGs are described by a kineticequation formed by a transport equation for the slowly-varying DOS u ( λ, x, t ) and the integral equation of state Figure 2. (a), (b) Propagation of a Tajiri-Watanabe breather with the spectral parameter η [1] = 0 .
05 + 1 . i inside a PeregrineBG. The space-time evolution shown in (b) represents an enlarged view of the one shown in (a). The white dashed line in(a) and (b) represents the trajectory of the “free” Tajiri-Watanabe breather propagating on a plane wave background with agroup velocity given by Eq. (11). The plot shown in (c) represents the spectral portrait associated with the numerical resultsshown in (a), (b). The vertical line between 0 and + i represents the branchcut associated with the plane wave background andthe blue point is the discrete eigenvalue η [1] associated with the Tajiri-Watanabe breather propagating in the P-BG. The 50spectral points characterizing the P-BG are densely placed around + i and they are shown in the inset plotted in (c). relating the gas’ velocity to the DOS. In this Section,we show that some predictions of the spectral theory ofBGs can be verified in simulations involving BGs thathave been numerically synthesized using the methodol-ogy described in Sec. II B. In Sec. III A, we provide thekey elements of spectral theory of BGs that are relevantfor the comparison between theoretical and numerical re-sults. In Sec. III B, we examine the collision between onetrial soliton and various single-component BGs. A. Analytical results from the spectral theory ofbreather gases
The nonlinear spectral theory of SGs and BGs for thefocusing 1D-NLSE developed in ref. [23] provides a fullset of equations characterizing the macroscopic spectraldynamics in a spatially nonhomogeneous BG.An important result of the theory is the so-called equa-tion of state which provides the mathematical expressionof the modification of the mean velocity of a “tracer”breather due to its interaction with other breathers inthe gas.The group velocity (in the ( x, t )-plane) of a first-orderbreather (TW) parametrized by the complex eigenvalue λ ≡ η (we shall use in this section this latter notation forthe spectral parameter to be consistent with notationsof ref. [23] and previous works on the spectral kinetictheory) is given by s ( η ) = − (cid:61) [ ηR ( η )] (cid:61) [ R ( η )] (11)where R ( z ) = (cid:112) z − δ with δ the endpoint of thebranchcut corresponding to the plane wave ( δ = i for the plane wave of unit amplitude considered in all thenumerical simulations reported in this paper). It is notdifficult to see that, if η ∈ i R \ [ − i, i ] (KM breather) then s ( η ) = 0, while if η ∈ ( − i, i ) (AB) then s ( η ) = ±∞ depending on the way the limit Re ( η ) → s ( η ) = s ( η ) + (cid:90) Λ + ∆( η, µ ) (cid:2) s ( η ) − s ( µ ) (cid:3) u ( µ ) | d µ | (12)where Λ + is the 2D compact support of the DOS u ( η )(defined earlier in Section II B) located in the upper halfplane C + of the complex spectral plane,∆( η, µ ) = 1 (cid:61) [ R ( η )] (cid:104) ln (cid:12)(cid:12)(cid:12) µ − ¯ ηµ − η (cid:12)(cid:12)(cid:12) + ln (cid:12)(cid:12)(cid:12) R ( η ) R ( µ ) + ηµ − δ R (¯ η ) R ( µ ) + ¯ ηµ − δ (cid:12)(cid:12)(cid:12)(cid:105) . (13)The integral term in Eq. (12) describes the modificationof the ‘tracer’ breather mean velocity in a gas due toits interaction with other breathers in the gas having aDOS specified by u . The spectral value η in (12) canbe taken outside Λ + —in that case formula (12) describesthe mean velocity of a “trial” or “test” TW breather withthe eigenvalue η propagating through a breather gas withDOS supported Λ + .The interaction kernel ∆( η, µ ) given by Eq. (13) de-scribes the position shift arising in a two-breather inter-action. We note that the two-breather interactions havebeen studied in [73], [49] using the IST, where different Figure 3. (a), (b) Propagation of a TW breather with the spectral parameter η [1] = 0 .
05 + 1 . i inside a Kuznetsov-Ma BG.The space-time evolution shown in (b) represents an enlarged view of the one shown in (a). The white dashed line in (a) and(b) represents the trajectory of the “free” TW breather propagating on a plane wave background with a group velocity givenby Eq. (11). The plot shown in (c) represents the spectral portrait associated with the numerical results shown in (a), (b).The vertical line between 0 and + i represents the branchcut associated with the plane wave background and the blue point isthe discrete eigenvalue η [1] associated with the TW breather propagating in the KM-BG. The 50 spectral points characterizingthe KM-BG are densely placed around η [2] = 1 . i and they are shown in the inset plotted in (c). forms of the expressions for the position shift were ob-tained. In the Appendix we demonstrate the equivalenceof the kernel ∆( η, µ ) given by (13) to the position shiftformula obtained for two-breather collisions in previousworks.For a two-component breather gas, the DOS is a su-perposition of two Dirac delta-functions centered at thecomplex spectral points η [ j ] ( j = 1 , u ( η ) = (cid:88) j =1 w [ j ] δ ( η − η [ j ] ) (14)where w [ j ] are the weights of the components. For theDOS specified by Eq. (14), Eq. (12) yields the followinglinear system for the gas’ component velocities s [ j ] ≡ s ( η [ j ] ) ( j = 1 , s [1] = s [1]0 + ∆ , w [2] ( s [1]0 − s [2]0 )1 − (∆ , w [2] + ∆ , w [1] ) s [2] = s [2]0 − ∆ , w [1] ( s [1]0 − s [2]0 )1 − (∆ , w [2] + ∆ , w [1] ) (15)where s [ j ]0 ≡ s ( η [ j ] ) ( j = 1 , j,k = ∆( η [ j ] , η [ k ] ).In the numerical simulations presented in Sec. III B,we will consider an even simpler situation where a singletrial breather parametrized by the eigenvalue η [1] inter-acts with a one-component breather-gas having its spec-tral distribution centered in η [2] . In such a limit w [1] → s [1] = s [1]0 − ∆ , w [2] s [2]0 − ∆ , w [2] .s [2] = s [2]0 . (16) The validity of Eqs. (16) in the context of the 1D-NLSE dynamics (1) will be verified for the P-BG, theKM-BG and the AB-BG in numerical simulations pre-sented in Sec. III B. As a matter of fact, formula (16)can be obtained directly from equation (12) by setting η = η [1] / ∈ Λ + (the trial breather eigenvalue), and using u ( µ ) = w [2] δ ( µ − η [2] ), s ( η ) = s [2]0 where η [2] ∈ Λ + . B. Interactions in one-component breather gases:Comparison between spectral theory and numericalsimulations
In the numerical simulations presented in this Sec-tion, a trial TW breather with the spectral parameter η = η [1] is propagated through various single-componentBGs having their DOS defined by u ( η ) = w [2] δ ( η − η [2] ).We define spectral parameter η [2] as η [2] = α i with α = 1for the P-BG, α > α < N = 50 spectral points randomly placed in a smallsquare region of width δ = 10 − centered around η [2] .The spectral parameter η [1] is chosen in such a way that (cid:60) ( η [1] ) > x − t ) plane, see Eq.(11).
1. Interactions in the Peregrine breather gas
Fig. 2 shows a trial Tajiri-Watanabe breather prop-agating through a P-BG. We observe that the trialbreather passes through the P-BG without change in itsgroup velocity. This confirms the theoretical result es-tablished in ref. [23] that the propagation of a trial TWbreather through a P-BG is ballistic. This result can beunderstood at the qualitative level by the fact that theinteraction cross section between the trial breather andthe individual Peregrine breathers composing the gas isso weak that the propagation of the trial breather is un-affected by the P-BG.
2. Interactions in the Kuznetsov-Ma breather gas
Fig. 3 shows a trial TW breather propagating througha KM-BG. Contrary to Fig. 2, the multiple interactionsbetween the trial breather and the KM breathers com-posing the KM-BG now significantly influences the prop-agation ot the trial breather, see Fig. 3(a) and 3(b) fora comparison between the trajectory of the free Tajiri-Watanabe breather (in white dashed lines) and the tra-jectory followed by the trial breather in the KM-BG. Asshown in Fig. 3(b), the trial breather acquires a signif-icant space shift each time that its trajectory intersectsthe trajectory of an individual KM breather composingthe BG. At the macroscopic scale, this produces a ve-locity change of the trial breather inside the KM-BG.This leads to a spatial shift ∆ X in the position of thetrial breather which is measurable when the trial breatheremerges from the KM-BG, see Fig. 3(a).For the KM-BG, Eq. (16) simplifies to s [1] = s [1]0 − ∆ , w [2] (17)given that s [2]0 = 0. Eq. (17) clearly shows that the groupvelocity of the trial Tajiri-Watanabe breather is increasedby a factor 1 / (1 − ∆ , w [2] ) due to the interaction withthe KM-BG.Note that the space shift ∆ X acquired by the trialbreather as a result of propagation inside the KM-BGsimply represents the product of the number N of in-terations (equivalently the number of breathers in theKM-BG) with the elementary space shift ∆ , inducedby each interaction: ∆ X = N ∆ , . This provides an al-ternative and straightforward way to check the validityof Eq. (17) which gives the group velocity of the trialbreather inside the KM-BG.A set of numerical simulations with different values ofthe spectral parameters η [1] and η [2] has been made tocheck the validity of the spectral theory. Different real-izations of the KM-BG have been made and the value of w [2] is determined from numerical simulations as the ra-tio between the selected number N of breathers in the gasover the spatial extension L of the gas: w [2] = N/L
Asshown in Fig. 4, we observe full quantitative agreementbetween the numerical experiment and the predictions ofthe spectral theory.
Figure 4. Quantitative verification of the spectral theory ofBGs introduced in ref. [23]. Comparison between numerics(red dots) and theory (dashed lines) for the effective velocity( s [1] ) of a trial breather ( η [1] ) propagating in a a KM-BG( η [2] ).
3. Interactions in the Akhmediev breather gas
The case of AB-BG is special and requires a separateconsideration, particularly because it has not been con-sidered in any detail in [23]. The AB is a “static” ob-ject, not localized in space, so it is not immediately ob-vious how to identify the key quantities u ( η ) and s ( η )for the AB-BG. A single AB is a limiting case of theTW breather where the soliton eigenvalue η [2] is placedwithin the branch cut [0 , i ] in the upper half plane, TheAB-BG is generally characterized by some distribution ofsoliton eigenvalues along the branch cut. Similar to theabove consideration of KM-BG, we consider the AB-BGwith soliton eigenvalues clustered around a given spectralpoint η [2] (and c.c.) to mimic a one-component gas.As we have already mentioned in Section III A the for-mula (11) for the group velocity of the TW breather im-plies | s ( η ) | → ∞ as η → η [2] , which is consistent with thedelocalized nature of the AB. On the other hand, it canbe shown using the results of ref. [23], that in the AB-BGlimit the DOS u ( η ) → v ( η ) = s ( η ) u ( η ) = O (1). This motivates the followingalternative form of the equation of state (12): s ( η ) = s ( η ) + (cid:90) Λ + ∆( η, µ ) (cid:2) s ( η ) s ( µ ) − (cid:3) v ( µ ) | d µ | , (18)which is more suitable for the characterization of the AB-BG interactions. Equation (18) was obtained from (12)by substituting u ( η ) = v ( η ) s ( η ) . Assuming Λ + to be a nar-row region surrounding the branch cut [0 , i ] and using | s ( µ ) | (cid:29) µ ∈ Λ + equation (18) to leading orderbecomes s ( η ) = s ( η ) − (cid:90) Λ + ∆( η, µ ) v ( µ ) | d µ | . (19) Figure 5. (a), (b) Propagation of a TW breather with the spectral parameter η = 0 .
06 + 1 . i inside a Akhmediev BG. Thespace-time evolution shown in (b) represents an enlarged view of the one shown in (a). The white dashed line in (a) and (b)represents the trajectory of the “free” TW breather propagating on a plane wave background with a group velocity given byEq. (11). The plot shown in (c) represents the spectral portrait associated with the numerical results shown in (a), (b). Thevertical line between 0 and + i represents the branch cut associated with the plane wave background and the blue point is thediscrete eigenvalue η associated with the TW breather propagating in the AB-BG. The 50 spectral points characterizing theKM-BG are densely placed around η [2] = 0 . i and they are shown in the inset plotted in (c). Equation (19) describes the modification of the veloc-ity of the TW breather with eigenvalue η propagatingthrough the AB-BG characterized by the spectral fluxdensity v ( µ ).An important property of ∆( η, µ ) given by (13) is that∆( η, µ ) + ∆( η, − ¯ µ ) = 0 when µ ∈ [0 , i ] , (20)that is, when µ is on the branch cut [0 , i ]. The secondvariable η can take any value in the upper half-plane.Equation (20) implies that ∆( η, µ ) takes opposite valueson the opposite sides of the branchcut.It can further be shown that in the case of a breathergas, whose spectral support Λ + is symmetric with re-spect to the branch cut [0 , i ], the function v ( η ) also takesopposite values on the opposite sides of [0 , i ]. Thus thespeed of the AB-BG s ( η ) from (18) does not depend onwhich side of the upper part of the branch cut [0 , i ] thedomain Λ + or its parts are situated.Let us now consider a one-component AB-BG with thespectral flux v ( η ) = w t δ ( η − η [2] ), where η [2] ∈ [0 , i ] and w t is a real constant weight. As a result, equation (19)assumes a simple form s ( η ) = s ( η ) − w t ∆( η, η [2] ) , (21)We note that the sign of w t , as was explained above,depends on the side of [0 , i ] but the sign of the product w t ∆ does not. Hence we have the general result s ( η ) − s ( η ) < w [2] → w [2] s [2]0 ≡ w t . This simple formal consideration, how-ever, does not provide the important information aboutthe sign of w t ∆. Fig. 5 shows a trial TW breather propagating througha AB-BG. Similar to Fig. 3, the propagation of the trialbreather is significantly influenced by the the multipleinteractions with the AB breathers composing the AB-BG, see Fig. 5(a) and 5(b). One can see that, in con-trast to the interaction of the trial TW breather with theKM-BG, the group velocity of the trial TW breather isreduced in the interaction with the AB-BG, in agreementwith Eq. (21). Indeed, the space shifts observed in Fig.3(a) and in Fig. 5(a) have opposite signs.Similar to the KM-BG interactions, a set of numericalsimulations with different values of the spectral parame-ters η [1] and η [2] has been made to check the validity ofequation (21). Different realizations of the AB-BG havebeen produced and the value of w t was determined fromnumerical simulations as the ratio between the selectednumber N of AB in the gas over the temporal extension T of the gas: w t = N/T
As shown in Fig. 6, we ob-serve full quantitative agreement between the numericalexperiment and the predictions of the spectral theory.
IV. CONCLUSIONS
We have developed a numerical algorithm of the ISTspectral synthesis of breather gases for the focusing 1D-NLS equation. The algorithm is based on the recur-sive Darboux transform scheme realized in high precisionarithmetics. Using this algorithm we have synthesizednumerically three types of “prototypical” breather gases:the Akhmediev, Kuznetsov-Ma and Peregrine gas.Using the developed spectral algorithm, the interactionproperties of breather gases, predicted by the kinetic the-ory of ref. [23] have been tested by propagating through0
Figure 6. Comparison between numerics (red dots) and the-ory (dashed lines) for the effective velocity ( s [1] ) of a trial TWbreather ( η [1] ) propagating in a a AB-BG ( η [2] ). them a ‘trial’ generic TW breather whose effective veloc-ity is strongly affected by the interaction with the gas. Inall cases the theoretically predicted effective mean veloc-ity of the trial breather propagating through a breathergas demonstrates excellent agreement with the results ofthe numerical simulations. The verification of the theory,despite the inevitable effects of modulational instabilitypresent in the 1D-NLSE dynamics, has been made possi-ble due to the whole numerical algorithm being based onthe spectral construction rather than direct simulationsof the 1D-NLSE equation.The quantitative verification of the kinetic theory ofbreather gases undertaken in this paper is an importantstep towards a better understanding of this type of aturbulent motion in integrable systems. We also believethat the ability to synthesize numerically BGs representsa step of importance towards the controlled laboratorygeneration of BGs, possibly following an approach similarto the one recently reported for hydrodynamic SGs [18].Finally the possibility to generate numerically breathersolutions of order N (cid:38)
10 paves the way for further worksdevoted to the investigation of the properties of localiza-tion in space and time of breather solutions of the 1D-NLSE of very high order [37, 49, 71].
ACKNOWLEDGMENTS
This work has been partially supported by the AgenceNationale de la Recherche through the I-SITE ULNE(ANR-16-IDEX-0004), the LABEX CEMPI (ANR-11-LABX-0007) and the Equipex Flux (ANR-11-EQPX- 0017), as well as by the Ministry of Higher Educationand Research, Hauts de France council and EuropeanRegional Development Fund (ERDF) through the CPERproject Photonics for Society (P4S), EPSRC grant (UK)EP/R00515X/2 (GE), NSF (USA) grant DMS 2009647(AT) and Dstl (UK) grant DSTLX-1000116851 (GR, GE,SR). GE, AT and GR thank the PhLAM laboratory atthe University of Lille for hospitality and partial financialsupport.
APPENDIX: POSITION SHIFT INTWO-BREATHER INTERACTIONS
The two-breather interactions have been studied inrefs. [73], [49] where the expressions for the phase andposition shifts in the interaction of two Tajiri-Watanabebreathers have been derived using the IST analysis. InSection III A of this paper the interaction kernel in theequation of state (12) for breather gas has been obtainedin the form (13). The natural interpretation of this in-teraction kernel, consistent with the previously studiedcases of KdV and NLS soliton gases, is the position shiftin a two-breather collision. However, the equivalence be-tween formula (13) and the expressions from [73], [49] isfar from being obvious. Here we establish this equiva-lence enabling one to extend the phenomenological inter-pretation of soliton gas kinetics [22] to breather gases.We consider the position shift expression from [73]∆ ¯ ξ = − ln( ξ ) / ( c − , cos α ) = ∆( λ , λ ) , (22)where ξ = d + − α − α ) + c − , c − , ) cos( α − α ) d + − α + α ) − c − , c − , ) cos( α + α )(23)with c ± ,j = z j ± q /z j λ j = ( ζ j − q /ζ j ) / d ± ,j = z j ± q /z j q = − iδ d + = d + , + d + , R ( λ j ) = ( ζ j + q /ζ j ) / ζ j = R ( λ j ) + λ j = iz j e iα j . (24)One can verify that substituting (24) in (13) and invokingthe identities | λ i | = (cid:0) d + ,i + 2 q cos α i (cid:1) / d + = (cid:18) z z + q z z (cid:19) (cid:18) z z + z z (cid:19) (cos 2 α + cos 2 α ) / α + α ) cos( α − α )(25)yields the phase shift expression (22). 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