Bound state soliton gas dynamics underlying the noise-induced modulational instability
Andrey Gelash, Dmitry Agafontsev, Vladimir Zakharov, Gennady El, Stephane Randoux, Pierre Suret
BBound state soliton gas dynamics underlying the noise-induced modulationalinstability
Andrey Gelash , , Dmitry Agafontsev , , Vladimir Zakharov , ,Gennady El , St´ephane Randoux , , and Pierre Suret , , ∗ Skolkovo Institute of Science and Technology, Moscow, 143026, Russia Novosibirsk State University, Novosibirsk, 630090, Russia P. P. Shirshov Institute of Oceanology, RAS, 117218, Moscow, Russia P. N. Lebedev Physical Institute, 53 Leninsky ave., 119991, Moscow, Russia Northumbria University, Newcastle upon Tyne, United Kingdom Laboratoire de Physique des Lasers, Atomes et Molecules,UMR-CNRS 8523, Universit´e de Lille, France and Centre d’Etudes et de Recherches Lasers et Applications (CERLA), 59655 Villeneuve d’Ascq, France ∗ We investigate theoretically the fundamental phenomenon of the spontaneous, noise-induced mod-ulational instability (MI) of a plane wave. The long-term statistical properties of the noise-inducedMI have been previously observed in experiments and in simulations but have not been explainedso far. In the framework of inverse scattering transform (IST), we propose a model of the asymp-totic stage of the noise-induced MI based on N -soliton solutions ( N -SS) of the integrable focus-ing one-dimensional nonlinear Schr¨odinger equation (1D-NLSE). These N -SS are bound states ofstrongly interacting solitons having a specific distribution of the IST eigenvalues together withrandom phases. We use a special approach to construct ensembles of multi-soliton solutions withstatistically large number of solitons N ∼ Integrable partial differential equations (PDE) suchas Sine-Gordon, Korteweg-de Vries (KdV) and the one-dimensional nonlinear Schr¨odinger equation (1D-NLSE)are considered as universal models of nonlinear physics[1]. They describe at leading order various nonlinear sys-tems and can be integrated using the Inverse ScatteringTransform (IST), often seen as a nonlinear analogue ofthe Fourier transform [2–6]. In the IST theory, the wavefield plays the role of a potential in a linear scatteringproblem associated with the nonlinear PDE. The tradi-tional IST theory deals with decaying potentials [6, 7],but constant non-zero boundary conditions at infinitycan also be considered [8]. An extension of the ISTmethod to the case of periodic boundary conditions isalso available in the framework of the so-called finite gaptheory (FGT) [9, 10].In contrast to deterministic initial conditions, thepropagation of random waves in integrable systems is anopen theoretical problem of significant applied interestdue to complexity of many real world nonlinear phenom-ena modeled by integrable equations. For this reason,
In-tegrable Turbulence (IT) has been recently introduced asa “new chapter of turbulence theory” by V. Zakharov [11]and is now an active theoretical and experimental fieldof research [12–16]. The central question in integrableturbulence is the evolution of the statistical propertiesof a random wave field in the course of its propagationthrough a nonlinear dispersive medium. In particular,optics and hydrodynamics provide very favorable settingsfor the investigation of integrable turbulence because the nonlinear propagation of one-dimensional waves in watertank or in an optical fiber are described at leading orderby the 1D-NLSE or KdV models [17–21].It has been realised in [22] that the development of thenoise-induced
Modulational Instability (MI, also knownas the Benjamin-Feir instability) arising in the focusingregime of the 1D-NLSE represents a prominent exampleof IT phenomena. This instability can be observed inmany physical systems such as deep water waves [23],BEC [24] or nonlinear optical waves [25]. In the tradi-tional formulation, the development of MI is seen as theamplification of an initially small sinusoidal perturbationof a plane wave – the condensate [10, 22]. In this case,the nonlinear stage of MI is described by exact solutionsof the 1D-NLSE – Akhmediev Breathers [26–29].When the small initial perturbation of the condensateis a random process, the numerical simulations of thefocusing 1D-NLSE show that the long-time evolution ischaracterised by a stationary single-point statistics whichis Gaussian despite the presence of random highly non-linear breather structures [12, 14, 30]. It has also beenshown recently that this long-time (stationary) statis-tics is characterized by a quasi-periodic structure of thespatial autocorrelation function of the wave field inten-sity [31].While all these remarkable features of IT have beenrecently demonstrated experimentally by using an opticalfiber loop [31], they are still not understood theoretically.Integrable turbulence can be approached from a com-pletely different perspective which is close to classical sta- a r X i v : . [ n li n . S I] J u l tistical mechanics. In 1971, V. Zakharov introduced theconcept of soliton gas as an infinite collection of interact-ing KdV solitons that are randomly distributed in space[32]. Originally introduced for the case of small densitydue to significant simplifications in the analytical treat-ment, the notion of soliton gas has been extended to gasesof finite density both for KdV and for the focusing 1D-NLSE [33]. Importantly, the macroscopic properties ofdense soliton gases are determined by pairwise collisionsof solitons accompanied by phase shifts that are accu-mulated at long time leading to significant correctionsto the average soliton velocities. The key role in thesoliton gas theory is played by the spectral (IST) distri-bution function which has the meaning of the density ofstates f ( λ, x, t ), so that f ( λ , x , t ) dλdx is the numberof solitons with the spectral parameter λ in the interval[ λ , λ + dλ ] found in the space interval [ x , x + dx ]. Forspatially non-uniform soliton gas the evolution of f isdescribed by a kinetic equation [32, 33].Given the above two approaches to IT one can natu-rally pose a question about a possibility of describing thedevelopment of modulational instability of a plane waveby considering the soliton gas dynamics for some specialspectral (IST) distribution. In fact, the idea to explainnonlinear stage of MI using soliton interactions has beenput forward as early as in 1972 by V. Zakharov and A.Shabat [2]. However, rather paradoxically, up to now,this possible link between soliton interactions and MI ofa plane wave has not been explored.In this paper, we provide a bridge between the twofundamental phenomena of nonlinear physics by show-ing that soliton gas dynamics explains the fundamentalfeatures of the nonlinear stage of the noise-induced mod-ulational instability of a plane wave. More precisely, wedemonstrate a remarkable agreement between the spec-tral (Fourier) and statistical properties of an unstableplane wave in the long-time evolution and those of asoliton gas representing a random infinite-soliton boundstate (i.e. the 1D-NLSE solution in which all solitons arestationary in an appropriate reference frame). The twokey ingredients in our analysis are (i) a special choice ofthe spectral (IST) distribution in the soliton gas and (ii)random phases of the so-called norming constants.We consider the focusing 1D-NLSE in the standarddimensionless form: i ∂ψ∂t + 12 ∂ ψ∂x + | ψ | ψ = 0 , (1)where ψ ( t, x ) represents the complex field, and x and t are space and time. The plane wave solution of Eq. (1)is ψ c ( t, x ) = A exp iA t , where A is the condensate am-plitude. Without loss of generality, here we assume that A = 1 . The classical formulation of the noise-inducedMI problem is to consider the initial condition composed of the condensate with some additional noise: ψ ( t = 0 , x ) = 1 + η ( x ) , (2)where η is the noise with (cid:104) η (cid:105) = 0 and (cid:112) (cid:104)| η | (cid:105) (cid:28) k ) = k (cid:112) − k / k m = √ IST spectrum ). In the general case, the IST spec-trum of spatially localized wave fields ψ ( x ) – with zeroboundary conditions – consists of the continuous and dis-crete components. A special class of solutions, the N-soliton solutions ( N -SS), exhibits only discrete spectrumconsisting of N discrete complex-valued eigenvalues λ n , n = 1 , ..., N , and complex coefficients C n (norming con-stants) defined for each λ n . The key result of IST theoryis that, while the wave field ψ exhibits a complex dynam-ics, the IST spectrum changes trivially in time [7]: ∀ n : λ n = const , C n ( t ) = C n (0) e − iλ n t . (3)The asymptotic evolution of the N-SS for t → ∞ gen-erally leads to a superposition of N moving solitons (1-soliton solutions). Each soliton corresponds to a point λ i of the discrete spectrum so that Im λ i is proportional tothe soliton’s amplitude and Re λ i – to its velocity. Thevalue C i determines the soliton phase and its position.There is a special class of N-SS’s with all Re λ i = 0,the so-called bound states. In the following, we consideran ensemble of N-SS’s with the eigenvalues located onthe imaginary axis and random phases in Eq. (3), i.e. C n (0) = | C n (0) | e iθ n , where θ n are uniformly distributedin [0 , π ). For N (cid:29)
1, we assume the following (Weyl’s)distribution of IST eigenvalues λ n = i β n : f ( β ) = β/ ( (cid:112) − β ) (4)We shall call the limit at N → ∞ of the describedrandom N -SS ensemble a bound state soliton gas .The fundamental conjecture proposed and studied inthis paper is that spectral (Fourier) and statistical prop-erties of the stationary state of the noise-induced MI canbe described by a bound state soliton gas with certainstatistical distribution of the IST spectrum consisting ofthe Weyl’s distribution (4) of discrete eigenvalues andrandom, uniform distribution of the phases θ n . The as-sumption of random phase in the long-term evolutionof a stochastic field is natural because the phase rota-tions − iλ n t for large t introduce an effective randomiza-tion of the phases. Random phases of norming constantsare proposed here to describe IT in the framework ofIST; similarly, the so-called “random phase approxima-tion” in wave turbulence theory corresponds to randomphases of the Fourier components of weakly dispersivewaves [37, 38].The motivation behind the statistical eigenvalue dis-tribution (4) in the 1D-NLSE soliton gas is the Bohr-Sommerfeld quantization rule for the discrete spectrumof the semi-classical Zakharov-Shabat scattering problemwith the potential in the form of a real-valued rectangu-lar box of unit amplitude and width L (cid:29) λ n = i β n = i (cid:115) − (cid:20) π ( n − L (cid:21) , n = 1 , , . . . , N, (5)where N = int[ L /π ]. Importantly, for a ”semi-classical”box, the contribution of the ”non-soliton” part of thefield, i.e. of the continuous IST spectrum, to the solutiondecays exponentially with L and so can be neglected [7].The continuum limit of Eq. (5) with N → ∞ , β n → β yields the Weyl’s distribution (4) for f ( β ) = L dn/dβ .The derivation of the general N -SS of the 1D-NLSEis a classical result of the IST theory [2]. However, thenumerical computation of N -solitons solutions of the fo-cusing 1D-NLSE with large N ∼
100 has been realizedfor the first time in 2018 [40]. Here, we use the approachdeveloped in [40] that is based on the specific implemen-tation of the so-called dressing method [41] combinedwith 100-digits arithmetics (see [40] for details). Inthe following, we shall compare the nonlinear stage ofMI with the dynamical and statistical properties of thebound state soliton gas built as a random ensemble of N -solitons solutions with N = 128 and the spectral dis-tribution (4). In the results reported in this paper, thediscretisation of the Weyl distribution (4) is taken at theBohr-Sommerfeld quantization points (5). Similar resultsare obtained when soliton eigenvalues are randomly dis-tributed using the probability function (4).We study 10 realisations of 128-SS with random uni-form distribution of soliton phase parameters θ n in theinterval [0 , π ). The density of the gas, i.e. the numberof soliton per unit length, plays a crucial role in the dy-namics. Our numerical investigations have shown thatthe higher the density is, the better is the agreement be-tween soliton gas and the stationary state of MI. In thedressing method, the density is empirically controlled by“space position parameters” x n and has its maximum(critical) density when all x n = 0 and all | C n | = 1. Thiscritical density, which we observe empirically, coincideswith the density of solitons corresponding to the Bohr-Sommerfeld quantization rule (5). Note that, when all x n = 0, the N -SS solution is symmetric; in order toavoid this artificial symmetry, we use a random uniformdistribution of space position parameters x n in a nar-row interval [ − , Figure 1:
Example of one realization of 128-SS withrandom soliton phases. (a) Intensity profile | ψ ( x, t = 0) | .(b) Soliton eigenvalues are computed from Eq. (5). An example of the bound state N-SS with N = 128is displayed in Fig. 1. We first compare qualitativelythe temporal evolution of this bound state soliton gasand the temporal evolution of an unstable plane waves(Fig. 2). We simulate the MI development using pseu-dospectral Runge-Kutta 4th-order method as describedin [12]. Periodic boundary conditions in a box of size L (cid:39)
570 are used and the initial conditions are given byEq. 2 with (cid:104)| η | (cid:105) = 10 − . In the spatio-temporal dynam-ics of the MI, one recognizes the emergence of the well-known structures resembling the Akhmediev-Breathers(Fig. 2.a). The spatio-temporal evolution of the boundstate N-SS is also computed by using numerical simula-tions of 1D-NLSE (Fig. 2.b). Remarkably, the featurescharacterizing the dynamics of the N -soliton and of thenoise-induced MI are qualitatively very similar. Notethat, having purely imaginary eigenvalues, the N-SS usedhere are bound states and the solitons do not separate atlong time.We now compare quantitatively the statistical proper-ties of soliton gas and of the nonlinear stage of MI. Inparticular, the long-term evolution of the noise-inducedMI is characterised by stationary values of the poten-tial and kinetic energy, kurtosis and also by stationaryshapes of the (Fourier) spectrum, the probability den-sity function (PDF) of the intensity I = | ψ | , and theautocorrelation function g (2) (see [12, 36]).The total energy (Hamiltonian) E of the wave field is Figure 2:
Numerical simulations of 1D-NLSE : Space-Time diagrams of | ψ ( x, t ) | . (a) Noise-induced Modula-tional Instability of a plane wave. (b) Dynamics of the ran-dom phase N -SS (the initial condition | ψ ( x, | is shown inthe Fig. 1.a)Figure 3: The evolution of ensemble averaged kinetic (cid:104) H l ( t ) (cid:105) and potential (cid:104) H nl ( t ) (cid:105) energies for the noise induced develop-ment of MI (black curves) and random phase 128-SS (redcurves). one of the infinite constants of motion of 1D-NLSE [7]: E = H l + H nl , H l = 12 1 L (cid:90) L/ − L/ | ψ x | dx,H nl = −
12 1 L (cid:90) L/ − L/ | ψ | dx. (6)In the case of MI, it has been shown that after some os-cillatory transient, the kinetic energy reaches a station-ary value H l = 0 . H nl = − (cid:104) H l (cid:105) and (cid:104) H nl (cid:105) (dashed lines in Fig. 3).Here the averaging (cid:104) . . . (cid:105) has been performed over 10 random phase realisations.In the following, we perform ensemble averaging, to-gether with temporal averaging, both for the noise- induced MI and for the N -SS. In the case of the con-densate, the temporal averaging is performed when thesystem is sufficiently close (by its statistical properties)to the asymptotic stationary state ( t ∈ [160 , N -SS, time averaging is started from the verybeginning of the system evolution. It is extremely impor-tant to note that the time-averaging is used here only forpractical reasons. Simulations made by averaging solelywith ensembles of realizations of random phases of thenorming constants provide the same results.The wave-action spectrum, S k ∝ (cid:104)| ψ k | (cid:105) , ψ k = 1 L (cid:90) L/ − L/ ψ e − ikx dx, (7)of the asymptotic state of the MI and of the considered128-SS soliton gas coincide with excellent accuracy, asdemonstrated in Fig. 4a. 4. Note that S k is renormalizedporportionnaly to the spatial extension of the field.Moreover, soliton gas and noise-induced MI exhibitnearly identical PDF P ( I ) of the field intensity I = | ψ | (Fig. 4.b). The PDF of the N -SS reproduces quanti-tatively the exponential distribution discovered earlieras the asymptotic characteristic of the unstable conden-sate [12, 31].It has been shown very recently in [31] that the longterm evolution of the MI is typified by an oscillatorystructure of the second order degree of coherence (au-tocorrelation of the intensity) g (2) ( x ) : g (2) ( x ) = (cid:10)(cid:82) L/ − L/ I ( y, t ) I ( y − x, t ) dy (cid:11)(cid:10)(cid:82) L/ − L/ I ( y, t ) dy (cid:11) . (8)As can be seen from the Figs. 4.c the N -SS reproduce ac-curately this remarkable oscillatory shape. Note that H l , H nl , the PDF and the g (2) functions of the N -SS werecomputed in the central part of the soliton gas. Moreprecisely, we used the region x ∈ [ − N -SS is uni-form and very close to unity, that allows us to mitigatethe edge effects.As we have just shown, the asymptotic stage of thenoise-induced MI is accurately modelled by a soliton gaswith a special distribution of the IST eigenvalues λ , co-inciding with the semi-classical distribution (4) for thediscrete spectrum of the box potential [2]. This dis-tribution has been recently shown to describe the den-sity of states in the bound state soliton gas at criticaldensity [43]. Our numerical investigations reveal thatspectral (Fourier) and statistical properties of soliton gasmade of stochastic N -solitons bound states are very sen-sitive to the exact distribution of the IST eigenvalues.To illustrate the latter point, we compare in Fig. 5the typical (Fourier) spectrum of the asymptotic stateof MI (identical to Fig. 4.a) with the spectrum of thebound state soliton gas made of a N -SS with N = 128, Figure 4: Comparison of ensemble averaged statistical char-acteristics of the asymptotic state of the MI development andrandom phase 128-SSs. (a) Wave action spectrum S k . (b)The PDF P ( I ). (c) Second order degree of coherence (auto-correlation function of intensity) g (2) ( x ) random phases and equidistant eigenvalues in the interval[ π/ − a ; π/ a ]: λ n = i β n = i (cid:20) π a nN (cid:21) , where a = 0 .
025 (9)In sharp contrast to the “Weyl” soliton gas studiedabove, the Fourier spectra of MI and of the soliton gashaving the eigenvalue distribution (9) strongly differ.The detailed investigation of different soliton gases isbeyond the scope of the paper; note however that thespectrum of soliton gas presented in Fig. 5 resemblesthe one of unstable cnoidal waves studied in [44]. Thisexample shows that the agreement between MI and asoliton gas (Fig. 5) is allowed by the careful choice ofthe density of states f ( λ ).In this paper we have demonstrated that the spectraland statistical properties of the asymptotic stage of MI Figure 5: Example of wave action spectrum for soliton gashaving equidistant distribution of λ . precisely coincide with the ones of some specific solitongas. This soliton gas can be constructed with exact N -soliton solutions of the 1D-NLSE having large values N and one specific distributions of IST eigenvalues com-puted in the semi-classical limit [2]. As it could be ex-pected, the long-term statistical state of MI correspondto a full stochastization of IST phases. While in this pa-per we concentrate on the asymptotic stage of the noise-induced MI, the proposed soliton gas framework may beuseful in the understanding of the randomisation of thephases in the early stage of the MI.We believe that our work opens a new promising direc-tion in the theory of integrable turbulence by establishinga link between the modulational instability and the soli-ton gas dynamics. One of the most challenging problemsis the rigorous derivation of the normal distribution forthe complex field ψ ( x, t ) that typifies the nonlinear stageof noise-induced MI.Our model is based on the well-known multi-solitonsolutions and can be generalised to a broad class of in-tegrable turbulence problems when the (random) wavefield is strongly nonlinear, so that the impact of non-solitonic content can be neglected in the asymptotic state( t → ∞ ). In this case, the general recipe to study asymp-totic state is to build N -soliton solutions with the distri-bution f ( λ ) of eigenvalues characterizing the field and random phases of the norming constants.There is one important remark to make. As is known,the nonlinear stage of the modulational instability in-duced by small harmonic perturbations is character-ized by the generation of Akhmediev breathers [26–29].The picture is more complicated if the initial perturba-tion is a random noise, leading to integrable turbulencewith various breather structures appearing only locally[12, 35, 45, 46]. The local appearance of breathers hasalso been demonstrated in the context of multi-solitoninteractions [47, 48]. Our approach can shed light on thepossible connection between soliton gases and breathergases [30, 33, 43].Note finally that the mechanism underlying the MIinduced by noise studied here is a priori different fromthe MI induced by localized perturbations [36, 49–54].The local perturbations are studied whithin the frame-work of IST with nonzero background conditions, andthe corresponding dynamics can be strongly influencedby the soliton content of the perturbation [49, 52–54]. Incontrast, whithin the proposed soliton model of a fullydeveloped MI, we assume that the random perturbationsof the condenstate (at t=0) only induce random phasesof the special bound state soliton gas at t → ∞ .As we demonstrate in this work, the IST formalism forthe wave fields with decaying boundary conditions canbe successfully applied to describe accurately the asymp-totic stationary state of the MI computed numericallyby using periodic boundary conditions [12]. For an inte-grable PDE the periodic boundary problem is a subjectof periodic IST technique also known as finite gap the-ory [9, 10]. An important mathematical problem for thefuture studies is to explain the link between spatially pe-riodic and localised descriptions of the MI in terms of theIST theory. Acknowledgments
Simulations were performed at the Novosibirsk Su-percomputer Center (NSU). This work has been par-tially supported by the Agence Nationale de la Recherchethrough the LABEX CEMPI project (ANR-11-LABX-0007) and by the Ministry of Higher Education and Re-search, Nord-Pas de Calais Regional Council and Eu-ropean Regional Development Fund (ERDF) throughthe Contrat de Projets Etat-R´egion (CPER Photonicsfor Society P4S). The work on the construction of themulti-soliton ensembles reported in the first half of thework was supported by the Russian Science Founda-tion (Grant No. 19-72-30028 to AG, DA and VZ). Thework of GE was partially supported by EPSRC grantEP/R00515X/1. The authors thank A. Tikan and F.Copie for fruitful discussions ∗ Corresponding author : [email protected][1] J. Yang,
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