Soliton spectra of random water waves in shallow basins
J. -P. Giovanangeli C. Kharif Y. A. Stepanyants
physics.flu-dyn







 

 

 

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        ^1) Aix Marseille Université,
        CNRS, Centrale
        Marseille, IRPHE UMR 7342, 13384, Marseille, France; 
        
        ^2) Faculty of Health, Engineering and Sciences, University of
        Southern Queensland, 
        Toowoomba, QLD, 4350, Australia and 
        ^3) Department of Applied Mathematics, Nizhny Novgorod State
        Technical University n.a. R.E. Alekseev, 
        Nizhny Novgorod,
        603950, Russia.
        
        
        Interpretation of random wave field on a shallow water in terms of
        Fourier spectra is not adequate, when wave amplitudes are not
        infinitesimally small. A nonlinearity of wave fields leads to the
        harmonic interactions and random variation of Fourier spectra. As
        has been shown by Osborne and his co-authors, a more adequate
        analysis can be performed in terms of nonlinear modes representing
        cnoidal waves; a spectrum of such modes remains unchanged even in
        the process of nonlinear mode interactions. Here we show that
        there is an alternative and more simple analysis of random wave
        fields on shallow water, which can be presented in terms of
        interacting Korteweg–de Vries solitons. The data processing of
        random wave field is developed on the basis of inverse scattering
        method. The soliton component obscured in a random wave field is
        determined and a corresponding distribution function of number of
        solitons on their amplitudes is constructed. The approach
        developed is illustrated by means of artificially generated
        quasi-random wave field and applied to the real data
        interpretation of wind waves generated in the laboratory wind
        tank. 
         Keywords and phrases: shallow water; wind waves; random wave
        field; wave spectra; solitons; numerical modelling; laboratory
        experiments 
         Mathematics Subject Classification: 76B15, 76B25, 35Q51,
        35Q53, 37K40
        
        
        
        
        Soliton spectra of random water waves in shallow basins
            J.-P. Giovanangeli^1), C. Kharif^1), and Y.A. Stepanyants^2, 3)Corresponding author: Yury.Stepanyants@usq.edu.au
        
            
        =====================================================================================================================
        
        
        
        § INTRODUCTION
        
        
        The traditional approach in the problem of wind wave study is
        based on the analysis of Fourier spectra and determination of
        their peculiarities. There is a vast number of papers both
        theoretical and experimental where this problem has been
        considered; it is impossible to list all of them here. Therefore,
        we refer only to the review chapter “Wind waves” by Zaslavsky
        and Monin in the book <cit.> where a reader can find key
        references in this field. A lot of interesting and useful
        information has been obtained about wind waves, and their analysis
        and interpretation have been implemented in terms of Fourier
        spectra.
        
        Meanwhile, the Fourier analysis of wind waves provides researchers
        with only some piece of objective information whereas many
        important features of wind waves remain hidden. One of the serious
        obstacles making the Fourier analysis ineffective in application
        to surface oceanic waves is the nonlinear character of such waves,
        whereas the Fourier analysis is a linear operation applicable to
        systems obeying the superposition principle.
        
        Osborne with co-authors (see, e. g., <cit.> and references therein) have developed the method of
        nonlinear spectral analysis of shallow water waves described by
        the Korteweg–de Vries (KdV) equation. Osborne's approach is based
        on the application of the inverse scattering method (ISM) to the
        analysis of random field data in a one-dimensional space domain
        with the periodic boundary conditions. The main idea of his
        approach was in the presentation of complex initial disturbance in
        terms of a set of elliptic functions (cnoidal waves). These
        functions can be considered as the nonlinear eigenmodes which are
        preserved in the process of wave field evolution in contrast to
        the linear sinusoidal eigenmodes. This means that a nonlinear wave
        spectrum calculated on the basis of these modes is invariant in
        time while a usual Fourier spectrum is variable due to nonlinear
        interactions between the different sinusoidal harmonics. An
        important feature of nonlinear eigenmodes is their reducibility to
        the sinusoidal eigenmodes in the case of small amplitudes. In
        other words, a nonlinear spectrum naturally reduces to the Fourier
        spectrum if the analyzed wave field is quasi-linear. However, the
        mathematical and numerical machinery used for calculation of
        nonlinear eigenmodes is not simple in contrast to the linear case
        and, apparently, it is impractical especially for the applied
        mariner engineers.
        
        Here we propose a very similar to Osborne's, but a bit different
        approach to the analysis of random water waves that is also based
        on the application of ISM. The essential feature of our approach
        is the interpretation of a random initial wave field in terms of
        an ensemble of solitons and quasi-linear ripples rather than the
        set of elliptic eigenmodes (a similar approach was recently
        realised in Ref. <cit.>). The idea is illustrated by an
        example of shallow water waves described by the classical KdV
        equation with the random initial data. If one takes some portion
        of random field data (which should be long enough), the number of
        solitons, their amplitudes, speeds, characteristic durations,
        etc., can be calculated then by means of ISM (e.g., by solving
        numerically the associated Sturm–Louiville problem) or by the
        direct numerical simulation of the corresponding KdV equation. In
        both cases, the numerical codes are currently very well developed
        and easily available. In the meantime, the knowledge of number of
        solitons obscured in the random wave field, their parameters and
        statistics is a matter of independent interest per se. We describe
        our approach below in detail and give some examples (preliminary
        results were reported at the conference OCEANS'13 MTS/IEEE in San
        Diego, USA <cit.>).
        
        Before we start, it is useful to remind that the soliton
        turbulence of rarified soliton ensembles in strongly integrable
        systems is trivial to certain extent – the distribution function
        of solitons is unchanged in time <cit.> (the
        definition of strongly and weakly integrable systems is given in
        <cit.>). This is a consequence of trivial character of
        soliton interactions in such systems. The solitons do not change
        their parameters after collisions, and only the paired collisions
        occur between them. However, the dynamics of a dense ensemble of
        solitons is more complicated and soliton turbulence is nontrivial
        even in the integrable nonlinear wave equations <cit.>. In particular, in Ref. <cit.> the
        quantitative criterion of the term “dense soliton gas” was
        introduced and was shown that the density of KdV soliton gas is
        bounded from above. The critical gas density, apparently, depends
        on soliton distribution function, which makes important the
        determination of such function in a concrete physical problem such
        as water wave turbulence. Numerical experiments confirming the
        developed theory in Ref. <cit.> for the particular model
        distribution functions were reported in <cit.>.
        
        The KdV model considered here is the typical example of strongly
        integrable system applicable to real physical systems. This makes
        topical the development of handy methods of extraction of soliton
        distribution function from natural complex fields. One of the
        experimental approaches to the solution of this practical problem
        has been considered for internal waves in Okhotsk Sea
        <cit.> and another example of processing of surface
        wave observational data was reported in Ref. <cit.>. We
        hope, this publication will stimulate further interest to this
        important problem.
        
        
        
        § THE KORTEWEG–DE VRIES MODEL AND DATA PROCESSING
        
        
        Let us assume that there is a data of recorded surface waves at
        some fixed point x_0 of a shallow-water basin. So that the
        elevation η of the water level at this point is the known
        function of time: η(x_0,t) = f(t) where f(t) is some random
        function. A typical example of random surface waves usually
        measured in shallow-water basin is shown in Fig. <ref>. Here
        these data were artificially produced by means of a computer using
        the random number generator just to illustrate the idea of our
        approach.
        
        
        For the description of further space evolution of data measured at
        the point x_0, the so-called, timelike KdV equation (TKdV Eq.)
        <cit.> is used:
        
        
        ∂η/∂ x + 1/c_0∂η/∂ t - αη∂η/∂ t - β∂^3η/∂ t^3 = 0, 
        
        where c_0 = √(gh);   α = 3/(2c_0h);
          β = h^2/(6c_0^3) with g being the acceleration due to
        gravity and h being an unperturbed water depth.
        
        In the process of evolution of an initial perturbation one can
        expect emergence of a number of solitons with different amplitudes
        and phases (e.g., time markers of soliton maxima in the wave
        record at a given point of observation). Soliton solution to the
        TKdV equation (<ref>) has the form:
        
        
        η(x,t) = A^2t - x/V/T, 
        
        where the velocity V and duration T are relate to the
        amplitude A:
        
        
        V = c_0/1 - c_0α A/3≈ c_0(1 +
        c_0α A/3),      T =
        √(12β/α A), 
        
        the approximate formula is valid for small amplitude solitons when
        c_0α A/3 ≪ 1.
        
        
        
        
        As it was mentioned in <cit.>, “it may not be possible to
        observe solitons in real space” because of numerous nonlinear
        interactions of solitons both with each other and with chaotic
        radiation background component. However, “it would be naive to
        conclude that solitons are not present or that their dynamics are
        not important” in the evolution of the initial perturbation.
        Inasmuch as solitons are very stable with respect to interaction
        with others wave perturbations and influence of external effects
        (such as viscosity, inhomogeneity, etc), it is a matter of
        interest to extract them from the irregular components of a wave
        field and to describe their statistical properties and
        contribution to the total wave energy.
        
        The solution of this problem can be done by the following way. Let
        us consider a very long portion of recorded measurement data of
        surface perturbation at any given point x_0. The characteristic
        duration of this portion T_p is assumed to be much greater than
        the typical soliton time scale T. Let us represent a
        perturbation with the help of some dimensionless function
        (t):
        
        
        η(0,t) = U(t/T_p), 
        
        where U is the characteristic wave “amplitude”, e.g., the
        maximum value of perturbation η(0,t) in the considered
        portion of data.
        
        By means of the transformation
        
        
        u = η/U,     ξ = -α U/T_px,     τ = 1/T_p(t -
        x/c_0), 
        
        Eq. (<ref>) and the corresponding “initial” perturbation
        (<ref>) can be reduced to the standard form <cit.> (the
        term “initial” used here as it is traditionally used in
        mathematics for the solution of Cauchy problem of differential
        equations, but in fact, the perturbation is given at the fixed
        spatial point, i.e., it is rather the boundary condition):
        
        
        u_ξ + uu_τ + 1/σ^2u_τττ = 0 
        
        
        u(0,τ) = (τ) 
        
        with one dimensionless parameter σ^2 known in the
        oceanography as the Ursel parameter and defined as
        
        
        σ^2 = α UT_p^2/β. 
        
        
        As it was mentioned above, we consider the case when the duration
        of the perturbation is long enough, so that σ^2 ≫ 1. In
        this case the number of solitons obscured in the “initial”
        perturbation is also very big in general, and it is reasonable to
        describe them by the distribution function f(A). This function
        determines the number of solitons dN within the interval (A, 
        A+dA) <cit.>
        
        dN = f(A)dA. 
        
        
        According to the theory developed in <cit.>, the distribution
        function can be calculated at large values of σ by means of
        the formula:
        
        
        f(A) =
        σ/4π√(3U)∫_Ldτ/√(2U(τ)
         - A), 
        
        where the interval of integration L is determined by the
        condition
        
        
        2U(τ) > A. 
        
        
        As follows from Eq. (<ref>), soliton amplitudes are
        distributed in the interval <cit.>
        
        0 < A < 2U[(τ)], 
        
        and their total number can be found from the formula
        
        
        N = ∫_0^∞f(A) dA =
        (σ/π√(6)) ∫_(τ) 
        >  0√((τ)) dτ. 
        
        
        Therefore for large σ the total number of solitons is
        determined only by those intervals of τ-axis where function
        (τ) is nonnegative!
        
        As well known, the KdV equation possesses an infinite number of
        conserved densities I_n<cit.>. One of them,
        
        
        I_2 = ∫_-∞^+∞η^2(x,t) dt, 
        
        is proportional to wave energy and therefore is of a special
        physical interest. For the sake of simplicity we will call this
        value simply the “energy”.
        
        The fraction of energy of a nonsoliton component of a perturbation
        to the total energy of “initial” perturbation can be determined
        by means of formulae (18.27), (18.28) and (19.8) in Ref.
        <cit.>:
        
        
        I_2^ns/I_2^tot = ∫_η(0,τ)  < 
        0η^2(0,τ) dτ/∫_0^T_pη^2(0,τ) dτ.
        
        
        The total energy of a soliton component in the wave field can be
        readily calculated, if soliton amplitudes are known:
        
        
        I_2^sol = 4√(β/3α)∑_k =
        1^NA_k^3/2 = 4h/3√(3g)∑_k =
        1^NA_k^3/2. 
        
        In the last expression the values of coefficients α and
        β for surface water waves were used (see above after
        Eq. (<ref>)).
        
        To illustrate the idea of this approach we consider below several
        examples of different “initial” perturbations.
        
        
        
         §.§ Sinusoidal perturbation and its modifications
        
        
          §.§.§ A sinusoidal perturbation
        
        
        Let us study the KdV Eq. (<ref>) with the periodical
        conditions in time. Let us set h = 1 m, then c_0 = 3.13 m/s,
        α = 0.479 s/m^2, β = 0.00543 s^3/m. Assume then
        that the perturbation has the form
        
        
        η(0,t) = Asinω t, 
        
        where A = 0.01 m, ω = 0.0628 1/s.
        
        The value of Ursel parameter for this perturbation is σ^2 =
        555 ≫σ_s^2, where σ_s^2 ≡ 12 is the value of
        Ursel parameter for a single soliton regardless of its amplitude
        and duration <cit.>.
        
        Figure <ref> shows this sinusoidal initial perturbation and
        the result of its evolution at the distance x = 1.1·10^3 m
        when it is completely disintegrated into the sequence of solitons.
        After that moment, solitons begin interact with each other and the
        wave field looks much more complex. In Fig. <ref> one can
        distinguish nine different solitons whose amplitudes are presented
        in Table 1. Soliton amplitudes were conditionally defined here as
        the difference between their maxima and the mean value of two
        nearest minima.
        
        
        
        Table 1. Amplitudes of solitons, A_k, emerging from the
        different initial perturbations. The fifth raw in the Table shows
        the relative difference RD in the soliton amplitudes of rows
        three and four in percents, this will be explained in subsection
        <ref>.
        
        
        
        
        wwwwwA_k →   A_1   A_2   A_3   A_4   A_5   A_6   A_7   A_8   A_9
        
        Initial pert. ↓   (mm)   (mm)   (mm)   (mm)   (mm)   (mm)   (mm)   (mm)   (mm) 
        
        Sin., periodic    29.39   26.7   23.72   20.37   16.38   11.61   6.625   2.876   0.792
        
        Sin., pulse    17.77   13.24   8.927   4.743   1.092   
            
           
            
        
        
        Half-sin., pulse    17.71   13.24   8.863   4.738   1.171   
            
           
            
        
        
        RD· 100%   0.34   0   0.723   0.1   -6.75   
            
           
            
        
        
        
        
        The total energy of the sinusoidal initial perturbation over a
        period T = 2π/ω can be easily evaluated
        
        
        I_2^tot = 1/2∫_0^Tη^2(0,t) dt =
        A^2/2∫_0^Tsin^2ω t dt =
        A^2T/4 = 2.5·10^-3 ^2 . 
        
        
        Similarly, the energy of the non-soliton component of the
        perturbation is (see Eq. (<ref>))
        
        
        I_2^ns = ∫_η(0,τ)  < 0η^2(0,τ) dτ
        = A^2/2∫_T/2^Tsin^2ω t dt =
        A^2T/8 = 1.25·10^-3 ^2 . 
        
        
        Thus, I_2^ns = I_2^tot/2, and the energy of all solitons in
        this case should be I_2^sol = I_2^tot/2 =
        1.25·10^-3 ^2. Let us calculate however
        this energy directly by means of Eq. (<ref>) using data of
        Table 1 for the soliton amplitudes 
        
        I_2^sol = 4h/3√(3g)∑_k = 1^9A_k^3/2
        = 4.922· 10^-3 ^2 . 
        
        
        This value about four times greater then the expected. The
        discrepancy can be explained by the incorrect counting of number
        of solitons, as well as their amplitudes. As has been shown in
        <cit.>, the problem of
        detection of solitons emerging from the harmonic perturbations is
        not so simple even within the KdV equation (disintegration of sine
        wave onto set of solitons was recently studied also within the
        Gardner equation <cit.>). Actual number
        of solitons is always greater then that at the instant of time
        when they appear from the initial perturbation for the first time
        in the ordered form. Their amplitudes are also different from
        those which are seen in Fig. <ref>. We will come back to this
        issue a bit later, and now let us consider a pulse-type initial
        perturbations defined on a compact support. 
        
        
          §.§.§ A pulse of sinusoidal profile
        
        
        Consider now a pulse-type initial perturbation having the shape of
        one period of sine with the same amplitude and characteristic
        duration as in the previous case (see dashed line in
        Fig. <ref>). In the process of evolution this perturbation
        disintegrates into a sequence of five solitons (see solid line in
        Fig. <ref>), whose amplitudes are indicated in the third row
        of Table 1. An intense oscillatory tail behind the solitons is
        also appeared; the head portion of this tail is shown in
        Fig. <ref>.
        
        
        
        By comparison of rows two and three of Table 1, one can see that
        the number of solitons and their amplitudes are absolutely
        different in the periodic and nonperiodic cases.
        
        As all these solitons are practically independent at the distance
        indicated in Fig. <ref>, their energies can be calculated
        independently. The calculation of cumulative energy of these five
        solitons yield: 
        
        I_2^sol = 4h/3√(3g)∑_k = 1^5A_k^3/2
        = 1.2531· 10^-3 ^2 . 
        
        
        This result already agrees quite well with the theoretical
        prediction.
        
        
        
          §.§.§ A half-sine pulse
        
        
        Let us consider now another pulse-type perturbation which contains
        only a half period of a sine-function of a positive polarity (see
        dashed line in Fig. <ref>).
        
        
        
        This perturbation disintegrates into the same number of solitons
        as in subsection <ref>. At the same distance x =
        6· 10^4 m, the soliton amplitudes are practically the same as
        in the previous case (cf. rows three and four in Table 1). The
        fifth raw in Table 1 shows the relative difference in the soliton
        amplitudes in percents, RD = (A_i' - A_i”)/A_i”, where A_i'
        and A_i” are the amplitudes of i-th solitons emerged from the
        sinusoidal and half-sine pulses respectively.
        
        As one can see from Table 1 and Figs. <ref> and <ref>,
        there is no much differences in the corresponding soliton
        amplitudes for the sinusoidal and half-sine pulses. In the next
        Subsection we will consider a simple analytical example to justify
        this numerical observation.
        
        
        
         §.§ Analytical solutions to the associated
        Schrödinger equation for the rectangular and meander-type
        pulses
        
        
        To understand better the regularity of distribution of amplitudes
        of solitons emerging from initial perturbations, let us consider
        two model initial perturbations: the rectangular pulse of positive
        polarity and amplitude U_0 (Fig. 5a) and meander-type pulse
        whose positive part coincides with the above rectangular pulse and
        negative part is the same but of opposite polarity (Fig. 5b).
        Assume that the duration of positive rectangular pulses are T.
        
        
        
        According to the inverse scattering method <cit.>, to study the evolution of the initial perturbation within
        the KdV Eq. (<ref>), one have to solve the complementary
        Schrödinger equation: 
        
        ψ”(ξ) + σ^2/6[(ξ) +
        E_n]ψ(ξ) = 0, 
        
        where ψ is the auxiliary function;  is the dimensionless
        function describing the shape of the initial perturbation (see
        above), it has the unit amplitude, i.e., _max = 1, and the
        unit characteristic duration in the dimensionless variables
        (<ref>). For the case a) in Fig. <ref>, = 1 at 0
        ≤ξ≤ 1 and zero beyond this interval. Similarly, for the
        case b) in Fig. <ref>, = 1 at 0 ≤ξ≤ 1, =
        -1 at 1 < ξ≤ 2 and zero beyond these intervals. Further,
        σ^2 is the Ursel parameter as defined in Eq. (<ref>)
        with U ≡ U_0 and T_p ≡ T; E_n < 0 with n = 1, 
        2 …, N are eigenvalues of the Schrödinger
        Eq. (<ref>). The amplitudes of emerging solitons, A_n, are
        related with the eigenvalues E_n by the simple relation: 
        A_n = -2U_0E_n. 
        
        
        The analytical solution to the Schrödinger Eq. (<ref>) with
        the rectangular potential function can be readily constructed (see
        e.g., <cit.>). Omitting simple, but tedious
        calculations, the result can be presented in the form of two
        transcendental equations determining eigenvalues E_n of the
        Schrödinger Eq. (<ref>):
        
        
        tan[σ^2√(β/24α(1 + E)) ] =
        √(-E/1 + E),-
        
        
        tan[σ^2√(β/24α(1 + E)) ] =
        -√(1 + E/-E). 
        
        
        In a similar manner the analytical solution can be found for the
        initial perturbation shown in Fig. <ref>b. The outcome is
        presented by the following transcendental equation:
        
            tan[σ^2√(β/6α(1 + E)) ] =
            -√(1 - E^2)(√(-E) + √(1 + E)) +
            √(1 + E)[√(-E(1 + E)) + 1 - E]/√(1 - E^2)(√(-E) - √(1 + E)) +
            [√(-E(1 - E)) - √(1 + E)(1 + E)],
        
        where =
        tanh(σ^2√(β/6α(1 -
        E))).
        
        
        Solutions of the transcendental Eqs. (<ref>)–(<ref>) can
        be presented graphically as shown in Figs. <ref>a) and
        <ref>b), where the series of tan-type curves represent the
        tangential functions in the left-hand side of these equations,
        while line 1 represent1 the right-hand side of Eq. (<ref>),
        line 2 represents the right-hand side of Eq. (<ref>), and two
        branches of line 3 represent the right-hand side of
        Eqs. (<ref>).
        
        The plots were generated for the same values of Ursel parameter
        σ^2 = 555, the water depth h = 1 m, and the amplitude of
        initial perturbations U_0 = 0.01 m were chosen the same as in
        Subsection 2.1 for the sinusoidal initial functions. In both cases
        of rectangular and meander-type pulses the number of roots of
        transcendetal equations (<ref>)–(<ref>) are the same, N
        = 8. Dashed vertical lines in each figure show positions of
        several first roots of corresponding equations. The amplitudes of
        solitons related to these eigenvalues as per Eq. (<ref>) are
        presented in Table 2. 
        
        
        Table 2. Amplitudes of solitons emerging from the rectangular and
        meander-type initial perturbations shown in Fig. <ref>.
        
        
        
        wwwwwA_k →   A_1   A_2   A_3   A_4   A_5   A_6   A_7   A_8
        
        Initial pert. ↓   (mm)   (mm)   (mm)   (mm)   (mm)   (mm)   (mm)   (mm) 
        
        Rectangular pulse    19.708   18.834   17.382   15.358   12.780   9.672   6.084   2.168
        
        Meander-type pulse    19.702   18.806   17.318   15.242   12.590   9.376   5.638   1.488
        RD· 100%   0.03   0.14   0.36   0.76   1.49   3.06   7.33   31.36
        
        
        
        The last row in the Table shows the relative difference in
        percents between the corresponding soliton amplitudes for the
        rectangular A^r_k and meander-type A^m_k perturbations: RD =
        (A^r_k - A^m_k)/A^r_k. As one can see from this Table, the
        difference between the corresponding amplitudes A_k is fairly
        small especially for the first largest solitons, and only for two
        last solitons of very small amplitudes the difference amounts for
        about 7% and 31% correspondingly. It can be readily shown that
        the larger the Ursel parameter, the greater the number of emerging
        solitons and the smaller the difference in theirs amplitudes.
        
        Thus, one can conclude that the asymptotic theory developed in
        <cit.> can provide a good basis for the calculation of
        statistical properties of solitons obscured in the random wave
        field. Apparently, the most energetic part of the soliton spectrum
        (a right wing of soliton distribution function at large
        amplitudes) is described fairly good while at small amplitudes the
        distribution function may be not quite correct.
        
        
         §.§ Cosine initial pulse
        
        
        Let us now consider why the number of solitons emerging in each
        period of a pure periodic perturbation (as well as their
        amplitudes) differs from those emerging from the pulse-type
        perturbations of the same shape, amplitude and duration (cf.
        Fig. <ref> with Figs. <ref> and <ref>)? The answer is,
        apparently, as follows: In the pure periodic case the zero level
        of the physical system is not determined clearly. The system
        admits  the minimum of the wave field as the zero level.
        Hence, a periodic sinusoidal perturbation can be treated as a
        periodic sequence of positive cosine-type pulses with respect to
        the minimum possible level. In a process of evolution, each pulse
        disintegrates into a number of solitons whose total amount is much
        greater than for sine-type perturbations considered above. New
        numerical simulation was carried out with this modified zero level
        for single cosine-type pulse as the initial perturbation; the
        result is shown in Fig. <ref>.
        
        There were clearly detected 11 solitons whose amplitudes are
        presented in Table 3. The amplitudes of solitons emerging from the
        periodic sinusoidal perturbation described above (see Subsection
        <ref>) are also shown in the same Table for the
        comparison. This time soliton amplitudes for the sinusoidal
        perturbation were calculated more thoroughly using trace method
        suggested in <cit.>. When the sinusoidal perturbation
        disintegrates for the first time into the sequence of solitons
        ordered by their amplitudes as shown in Fig. <ref>, the
        observer can not see all solitons because some of them are still
        obscured. We followed up for the development of the wave field
        after the distance x = 1.1· 10^4 m which corresponds to
        Fig. <ref> and discovered that in the process of soliton
        interactions some more solitons of small amplitudes appear. So
        that the total number of solitons in this case was exactly the
        same as for the cosine initial pulse, N = 11. Their amplitudes
        were measured with respect to the minima of the initial sinusoidal
        perturbation. As one can see, the relative difference in percents
        between the corresponding soliton amplitudes for the cosine
        initial pulse and periodic sinusoidal perturbation, RD = (A^c_k -
        A^s_k)/A^c_k is not too big now and not exceeds 17%. (see the
        last row of Table 3). 
        
        
        Table 3. Amplitudes of solitons, A_k, emerging from the cosine
        initial pulse and from the periodic sinusoidal perturbation.
        
        
        
        
        wwwwwA_k →   A_1   A_2   A_3   A_4   A_5   A_6   A_7   A_8   A_9   A_10   A_11
        
        Initial pert. ↓   (mm)   (mm)   (mm)   (mm)   (mm)   (mm)   (mm)   (mm)   (mm)   (mm)   (mm) 
        
        Cos., pulse    37.75   33.25   28.89   24.68   20.63   16.77   13.09   9.636   6.422   3.49   1.15
         Sin., periodic    33.70   30.39   26.92   23.36   19.60   15.54   11.43   8.04   5.66   -   -
        
        RD· 100%   10.73   8.60   6.82   5.35   4.98   7.35   12.65   16.60   11.82   -   -
        
        
        
        Thus, if the zero level of the considered physical system is known
        and the perturbation eventually vanishes at the infinity then the
        statistics of obscured solitons is determined by positive humps of
        the perturbation with respect to this zero level. Namely such a
        situation takes place in the case of hydrophysical measurements in
        laboratory (water tanks) or in natural marine conditions. But if
        the perturbation is periodic in principle, then solitons, their
        numbers and amplitudes are determined by each hump of the
        perturbation with respect to the total minimum of the initial
        perturbation.
        
        
        
         §.§ Data processing. A model example.
        
        
          §.§.§ A model spectrum and the range of its validity
        
        
        Let us apply now the developed approach to the random perturbation
        artificially generated and presented in Fig. <ref>. This
        perturbation was obtained by means of inverse Fourier transform of
        a series of 128 harmonics having random phases in the interval [0,
        2π] and amplitudes distributed in accordance with the formula
        
        
        S(ω) = S_0(ω + ω_0)^-2tanhω/δω, 
        
        where S_0 = 2· 10^-4 m, ω_0 = 2π·10^-2
        s^-1, and δω = 2.5· 10^-2 s^-1.
        
        
        
        For the construction of quasi-random wave field it was taken only
        the low-frequency and most energetic part of this spectrum, 0 ≤ω≤ω_lim, where ω_lim = 0.767 s^-1
        (see vertical dashed line in Fig. <ref>). The limiting
        frequency, ω_lim, was chosen for the following reasons.
        The dispersion relation for infinitesimal perturbations within the
        TKdV Eq. (<ref>) is
        
         k = ω/c_0 + h^2/6c_0^3ω^3  =
        ω/c_0(1 + h^2/6c_0^2ω^2), 
        
        where ω is the frequency of a sinusoidal wave, and k is
        the wave number, η∼ e^i(ω t - kx).
        
        As is well known, this dispersion relation represents an
        approximation of a real dispersion relation of a physical system,
        e.g., water or plasma waves, when ω, k → 0. The range of
        validity of the dispersion relation (<ref>) is restricted by
        the requirement that the second term in brackets is small in
        comparison with one (see, e.g., <cit.>). This
        condition gives
        
        
        ω≪ω_cr≡c_0/h√(6) = √(6g/h). 
        
        
        In our case ω_cr≈ 7.67 s^-1. To satisfy
        condition (<ref>), we set ω_lim = 0.1ω_cr =
        0.767 s^-1.
        
        
        
          §.§.§ Data processing of the model initial perturbation
        
        
        Let us consider now a quasi-random wave field shown in
        Fig. <ref> and apply the approach developed above. Each
        positive hump enumerated in the figure can be studied separately
        by means of the numerical code for the TKdV Eq. (<ref>). The
        numerical code was based on the explicit finite-difference scheme
        of a second-order accuracy both on spatial and temporal variables
        <cit.>. The theoretical analysis shows that used
        central-difference scheme is conditionally stable provided that
        Δ x ∼Δ t^3, where Δ x and Δ t are
        the spatial and temporal mesh steps. The code works very
        effectively and fast so that the result of pulse fission on
        solitons was obtained very quickly. A typical picture of
        disintegration of one of the pulses (pulse No 5 in Fig. <ref>)
        is shown in Fig. <ref>.
        
        
        The number of emerged solitons and their parameters can be easily
        calculated. To avoid errors in determination of soliton
        parameters, the numerical calculations were conducted until each
        soliton was sufficiently separated from others, so that their
        fields were not overlapped in the vicinity of their maxima. This
        procedure was carried out for each pulse shown in Fig. <ref>).
        As the result, it was obtained a large number of solitons of
        different amplitudes, their total number for all pulses was N_s =
        73 which is enough for the illustrative purposes. The solitons
        were collected into several groups (15 groups in total) according
        to their amplitudes. This allowed us to construct a histogram of
        numbers of solitons versus their amplitudes. The histogram can be
        considered as a model of the distribution function f(A) of
        density of number of solitons on amplitudes (see Eq. (<ref>)).
        The histogram obtained and the model distribution function for the
        considered illustrative example are shown in Fig. <ref>. Frame
        b) in the figure shows the cumulative distribution function, i.e.,
        the total number of solitons whose amplitudes are not greater than
        the given value.
        
        
        § LABORATORY EXPERIMENT WITH WIND WAVES ON SHALLOW WATER
        
        
        The theory developed above was applied to the data of laboratory
        experiments with wind wave generation. Series of experiments were
        carried out in the Luminy (Marseilles) small tank made of
        plexiglass and having the following sizes (length × width
        × height): 865 cm × 64 cm × 50 cm. The water
        depth in the tank in different experiments ranged from 1 cm to 8
        cm: h = 1, 2, 3, 4, 6, and 8 cm. Surface waves were
        generated by a wind blowing up over the water surface with the
        different mean velocities: V_w = 5.29,   6.45, 8.62 and 13.24
        m/s. The ventilator producing an air flow was installed at one of
        the ends of the tank. The blower width was the same as the width
        of the tank, 64 cm, but its height was 31 cm above the water
        level. The tank was covered above the blower by a plexiglass lid.
        At the opposite end of the tank it was placed a wave absorber to
        exclude reflected waves (an inclined bottom causing surface wave
        breaking). Two sensitive electric probes recording water level
        were placed at the distances 100 cm and 300 cm from the
        ventilator. The probes were thoroughly calibrated before each
        experiment; their sensitivities were equal and amounted s = 0.61
        V/cm.
        
        Below we present the analysis of only one of the series of
        experiments with the water depth h = 1 cm and wind velocity V_w
        = 5.29 m/s. Other experimental series were analysed in a similar
        way. We have to make a reservation in advance that the
        experimental data with wind waves are not perfect for the
        illustration of suggested approach. Wind waves generated by
        permanently blowing wind is an active system, i.e., the system
        with the permanent energy pumping at each point of water surface.
        Moreover, a distributed external force due to wind is not a
        constant, it varies from some maximum value near the ventilator to
        some small value at the opposite end of the tank. This results in
        the different soliton distribution functions measured at two
        different distances from the ventilator. A small water viscosity
        can also affect the soliton distribution function.
        
        
        Another difficulty with surface waves generated by wind is the
        effective generation of high frequency Fourier components, so that
        the essential portion of wave energy is contained in that part of
        Fourier spectrum which is beyond the range of validity of KdV
        equation. Therefore we were forced to restrict our analysis by
        only the low-frequency components of the wave spectrum. Figure
        <ref> presents the Fourier spectrum of wind waves recorded at
        two distances from the ventilator as indicated above. The critical
        frequency at h = 1 cm is ω_cr = 76.7 s^-1, and to
        satisfy the condition (<ref>) we cut the Fourier spectra of
        wind waves at ω_lim = 13.4 s^-1 (see dashed vertical
        line in Fig. <ref>) and ignored the high frequency portions of
        spectra above ω_lim.
        
        The surface perturbation was reconstructed by means of the inverse
        Fourier transform on the basis of extracted portions of wave
        spectra in the range 0 < ω≤ω_lim). The fragments
        of 60-second duration records corresponding to two spatial points
        of measurements are shown in Fig. <ref>.
        
        
        
        As it is clearly seen from the comparison of two data records
        presented in frames a) and b), there is the essential difference
        between them, especially in the intensity of wave fields. This can
        be explained by the effect of a fetch on wind wave generation.
        Therefore, one can expect that the statistics of obscured solitons
        in the record of frame b) is essentially richer than in frame a).
        
        The recorded data shown in Fig. <ref> were used as the input
        data for the TKdV equation (<ref>). In the statistically
        equilibrium state each 60-second portion of recorded data is
        equivalent to the same portion taken at a different time,
        therefore one can expect that the number of solitons obscured in
        each portion of data is the same in average and their distribution
        function is invariant with respect to time shift. This was
        confirmed in the data processing.
        
        The TKdV equation (<ref>) was solved numerically using the
        recorded data of 60 sec duration from the total time interval of
        208 sec. After a while solitons emerged from the quasi-random
        data, and their amplitudes were easily determined with the help of
        a special subroutine. This allowed us to determine the histogram
        of soliton numbers in the each particular interval of amplitudes
        A + Δ A; this is the analog of a differential distribution
        function. On the basis of this function we determined also the
        integral (cumulative) distribution function – the total number of
        solitons with the amplitudes less than A normalised by the total
        number of all solitons. Figure <ref> demonstrates the
        histogram of soliton numbers versus amplitudes for the time series
        shown in Fig. <ref>. The experimental data can be approximated
        by the Poisson distribution function P(n) =
        λ^ne^-λ/n!, where the parameter λ = 3.85
        for the histogram shown in frame (a) and λ = 4.94 for the
        histogram shown in frame (b).
        
        
        The corresponding integral distribution functions for the
        experimental data of Fig. <ref> are shown in Fig. <ref>
        (lines 1) together with the approximative Poisson integral
        functions with the same parameters as in Fig. <ref>. The total
        number of solitons emerged from the wave field shown in
        Fig. <ref>a) was 60, and emerged from the wave field shown in
        Fig. <ref>b) was 86. As expected, the time series recorded
        closer to the ventilator (Fig. <ref>a) contained less number
        of solitons than the time series recorded further from the
        ventilator (Fig. <ref>b). In the latter case the wave field
        was much better developed due to the influence of wave fetch.
        
        If we assume that all 60 solitons in the time series shown if
        Fig. <ref>a) are uniformly distributed in the time interval of
        60 s, then we obtain that the time interval per each soliton is
        Δ T_1 = 1 s. As follows from the histogram shown in
        Fig. <ref>a) the maximal number of solitons have amplitudes
        A_m1 = 0.02 cm and the duration T_s1 = 0.026 s. Therefore
        Δ T_1/T_s1≈ 3.85. The similar estimates for the
        time series shown if Fig. <ref>b) give the time interval per
        each soliton Δ T_2 ≈ 0.7 s. As follows from the
        histogram shown in Fig. <ref>b) the maximal number of solitons
        in this time series have amplitudes A_m2 = 0.05 cm and the
        duration T_s2 = 0.165 s. Therefore Δ T_2/T_s2≈
        4.2. Thus, we see that in both cases the “soliton gas” is very
        dense (cf. <cit.>). The fragments of numerical
        calculations presented in Fig. <ref> illustrate the soliton
        gas density in both time series.
        
        
        
        § CONCLUSION
        
        
        To analyze long random time series of water waves in shallow
        basins we have proposed an approach which differs from the
        traditionally used Fourier analysis. Our approach is based on the
        extraction of obscured solitons from the complex wave fields and
        construction of histograms of solitons at different points of
        observation. The histograms can be considered as experimental
        counterpart of distribution functions of number of solitons on
        their amplitudes. According to the theoretical conception, a
        soliton component of a wave field in the well-developed nonlinear
        perturbations should dominate. As is well known, the number and
        individual parameters of solitons preserve in the conservative
        statistically homogeneous systems <cit.>,
        therefore the distribution function (or histograms of solitons) is
        the same at different points of observation if the dissipative
        factors (i.e., viscosity or external sources of energy) are
        negligible. In contrast to that the Fourier spectrum changes due
        to nonlinearity.
        
        Our approach is in line with the contemporary development of the
        theory of strong turbulence in the integrable or near-integrable
        systems <cit.>. Experimentally constructed
        distribution function can be used for the determination of degree
        of soliton gas density – how far the density is from the critical
        value as defined in Refs. <cit.>. Data
        processing of laboratory experiments presented in our paper
        supplement the data processing of field experiments reported in
        Ref. <cit.>.
        
        A small dissipation can cause a gradual decay of soliton
        histograms and their distortion, in general. The histogram decay
        and its distortion depend on the specific type of dissipation;
        this problem has not been studied yet, although the decay of
        individual solitons under the influence of various types of
        dissipation has been investigated for the KdV <cit.>
        and Benjamin–Ono <cit.> solitons, as well as for the
        Kadomtsev–Petviashvili lumps <cit.>.
        
        Our approach can provide some additional valuable information
        about the energy distribution in natural wave fields such as the
        relationship between the soliton and nonsoliton components of the
        perturbation, and may indicate on the existence and intensity of
        external sources or sinks of energy.
        
        To determine the soliton number and amplitudes from the random
        time series, we applied direct numerical modelling for the
        evolution of initial data within the framework of the TKdV
        equation (<ref>). The existing numerical codes (see, e.g.,
        <cit.>) allow us to obtain the results fairly quickly in the
        form convenient to further analysis. However, it is not the only
        method which can be applied; the numerical solution of the
        eigenvalue problem (<ref>) can also be convenient and useful.
        Our experience with the application of the approach developed here
        shows that there is no problem with the determination of number
        and amplitudes of intense solitons, i.e., solitons of big and
        moderate amplitudes. However, it takes more efforts to determine
        parameters of solitons whose amplitudes are very small. In
        general, an uncertainty in the determination of parameters of
        solitons of very small amplitudes is higher than of moderate and
        big solitons. Meanwhile, the model example of subsection
        <ref> shown in Fig. <ref>, as well as the
        experimental laboratory data shown in Fig. <ref> demonstarte
        that the number of such small-amplitude solitons may be relatively
        big.
        
        As has been mentioned, the soliton distribution function remains
        unchanged in the integrable systems. However, the wave field
        randomly fluctuates in the process of evolution. This leads to the
        random fluctuations of local wave extrema. As has been shown in
        Ref. <cit.>, the distribution function of wave extrema
        varies with time even when the wave field consists of solitons
        only. This can be of interest from the viewpoint of physical
        applications, but beyond the scope of this paper.
        
        In conclusion, we emphasize that the approach developed here is
        applicable to the KdV-type systems, e.g., shallow-water waves
        (see, for example, Ref. <cit.> where the turbulence of
        soliton gas was studied both within the integrable KdV and
        non-integrable KdV-BBM equations). Its generalization to
        deep-water waves described by the Benjamin–Ono or nonlinear
        Shrödinger equation is the interesting and challenging problem.
        
         Acknowledgments. This work was initiated while one of the
        authors (Y.S.) was the invited visitor at the Laboratoire
        IRPHE–IOA, Marseille, France several years ago. Y. S. is grateful
        to CNRS, France for the invitation and to the Laboratory staff for
        the hospitality. Y.S. also acknowledges the funding of this study
        from the State task program in the sphere of scientific activity
        of the Ministry of Education and Science of the Russian Federation
        (Project No. 5.1246.2017/4.6) and grant of the President of the
        Russian Federation for state support of leading scientific schools
        of the Russian Federation (NSH-2685.2018.5). The authors are
        thankful to Efim Pelinovsky for useful discussions and valuable
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