Soliton spectra of random water waves in shallow basins
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
Keywords and phrases: shallow water; wind waves; random wave
field; wave spectra; solitons; numerical modelling; laboratory
Mathematics Subject Classification: 76B15, 76B25, 35Q51, 35Q53, 37K40
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 KamMon () 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., Osb1 (); Osb2 (); OSBC (); SBOP (); Osborne () 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. Costa ()). 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 GKRS-13 ()).
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 Zakh-71 (); Zakh-09 () (the definition of strongly and weakly integrable systems is given in Zakh-09 ()). 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 El-03 (); ElKam-05 (); El-16 (). In particular, in Ref. El-16 () 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. El-16 () for the particular model distribution functions were reported in Carbone-16 ().
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 Nagovitsyn () and another example of processing of surface wave observational data was reported in Ref. Costa (). We hope, this publication will stimulate further interest to this important problem.
Ii The Korteweg–de Vries model and data processing
Let us assume that there is a data of recorded surface waves at some fixed point of a shallow-water basin. So that the elevation of the water level at this point is the known function of time: where is some random function. A typical example of random surface waves usually measured in shallow-water basin is shown in Fig. 1. 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 , the so-called, timelike KdV equation (TKdV Eq.) OSBC () is used:
where with being the acceleration due to gravity and 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 (1) has the form:
where the velocity and duration are relate to the amplitude :
the approximate formula is valid for small amplitude solitons when .
As it was mentioned in OSBC (), “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 . The characteristic duration of this portion is assumed to be much greater than the typical soliton time scale . Let us represent a perturbation with the help of some dimensionless function :
where is the characteristic wave “amplitude”, e.g., the maximum value of perturbation in the considered portion of data.
By means of the transformation
Eq. (1) and the corresponding “initial” perturbation (4) can be reduced to the standard form Karp () (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):
with one dimensionless parameter known in the oceanography as the Ursel parameter and defined as
As it was mentioned above, we consider the case when the duration of the perturbation is long enough, so that . 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 . This function determines the number of solitons within the interval () Karp ()
According to the theory developed in Karp (), the distribution function can be calculated at large values of by means of the formula:
where the interval of integration is determined by the condition
and their total number can be found from the formula
Therefore for large the total number of solitons is determined only by those intervals of -axis where function is nonnegative!
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. Karp ():
The total energy of a soliton component in the wave field can be readily calculated, if soliton amplitudes are known:
In the last expression the values of coefficients and for surface water waves were used (see above after Eq. (1)).
To illustrate the idea of this approach we consider below several examples of different “initial” perturbations.
ii.1 Sinusoidal perturbation and its modifications
ii.1.1 A sinusoidal perturbation
Let us study the KdV Eq. (1) with the periodical conditions in time. Let us set m, then m/s, s/m, s/m. Assume then that the perturbation has the form
where m, 1/s.
The value of Ursel parameter for this perturbation is , where is the value of Ursel parameter for a single soliton regardless of its amplitude and duration Karp ().
Figure 2 shows this sinusoidal initial perturbation and the result of its evolution at the distance 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. 2 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, , emerging from the different initial perturbations. The fifth raw in the Table shows the relative difference in the soliton amplitudes of rows three and four in percents, this will be explained in subsection II.1.3.
The total energy of the sinusoidal initial perturbation over a period can be easily evaluated
Similarly, the energy of the non-soliton component of the perturbation is (see Eq. (15))
Thus, , and the energy of all solitons in this case should be . Let us calculate however this energy directly by means of Eq. (16) using data of Table 1 for the soliton amplitudes
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
SalMauEng (); Salupere-02 (); Salupere-03 (), 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 Kurkina-16A (); Kurkina-16B ()). 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. 2. 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.
ii.1.2 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. 3). In the process of evolution this perturbation disintegrates into a sequence of five solitons (see solid line in Fig. 3), 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. 3.
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. 3, their energies can be calculated independently. The calculation of cumulative energy of these five solitons yield:
This result already agrees quite well with the theoretical prediction.
ii.1.3 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. 4).
This perturbation disintegrates into the same number of solitons as in subsection II.1.2. At the same distance 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, , where and are the amplitudes of i-th solitons emerged from the sinusoidal and half-sine pulses respectively.
ii.2 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 (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 .
According to the inverse scattering method Whth (); Karp (); AblSeg (), to study the evolution of the initial perturbation within
the KdV Eq. (1), one have to solve the complementary
where is the auxiliary function; is the dimensionless function describing the shape of the initial perturbation (see above), it has the unit amplitude, i.e., , and the unit characteristic duration in the dimensionless variables (7). For the case a) in Fig. 5, at and zero beyond this interval. Similarly, for the case b) in Fig. 5, at , at and zero beyond these intervals. Further, is the Ursel parameter as defined in Eq. (10) with and ; with are eigenvalues of the Schrödinger Eq. (22). The amplitudes of emerging solitons, , are related with the eigenvalues by the simple relation:
The analytical solution to the Schrödinger Eq. (22) with the rectangular potential function can be readily constructed (see e.g., LandLif (); Flugge ()). Omitting simple, but tedious calculations, the result can be presented in the form of two transcendental equations determining eigenvalues of the Schrödinger Eq. (22):
In a similar manner the analytical solution can be found for the initial perturbation shown in Fig. 5b. The outcome is presented by the following transcendental equation:
Solutions of the transcendental Eqs. (24)–(26) can be presented graphically as shown in Figs. 6a) and 6b), 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. (24), line 2 represents the right-hand side of Eq. (25), and two branches of line 3 represent the right-hand side of Eqs. (26).
The plots were generated for the same values of Ursel parameter
, the water depth m, and the amplitude of
initial perturbations 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 (24)–(26) are the same, . 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. (23) are
presented in Table 2.
Table 2. Amplitudes of solitons emerging from the rectangular and meander-type initial perturbations shown in Fig. 5.
The last row in the Table shows the relative difference in percents between the corresponding soliton amplitudes for the rectangular and meander-type perturbations: . As one can see from this Table, the difference between the corresponding amplitudes 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 Karp () 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.
ii.3 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. 2 with Figs. 3 and 4)? 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. 7.
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
II.1.1) 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 SalMauEng (). When the sinusoidal perturbation
disintegrates for the first time into the sequence of solitons
ordered by their amplitudes as shown in Fig. 2, 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 m which corresponds to
Fig. 2 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, . 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, is not too big now and not exceeds 17%. (see the
last row of Table 3).
Table 3. Amplitudes of solitons, , emerging from the cosine initial pulse and from the periodic sinusoidal perturbation.
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.
ii.4 Data processing. A model example.
ii.4.1 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. 1. This perturbation was obtained by means of inverse Fourier transform of a series of 128 harmonics having random phases in the interval [0, ] and amplitudes distributed in accordance with the formula
where m, s, and s.
For the construction of quasi-random wave field it was taken only the low-frequency and most energetic part of this spectrum, , where s (see vertical dashed line in Fig. 8). The limiting frequency, , was chosen for the following reasons. The dispersion relation for infinitesimal perturbations within the TKdV Eq. (1) is
where is the frequency of a sinusoidal wave, and is the wave number, .
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 . The range of validity of the dispersion relation (28) is restricted by the requirement that the second term in brackets is small in comparison with one (see, e.g., AblSeg (); Karp (); Whth ()). This condition gives
In our case s. To satisfy condition (29), we set s.
ii.4.2 Data processing of the model initial perturbation
Let us consider now a quasi-random wave field shown in Fig. 1 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. (1). The numerical code was based on the explicit finite-difference scheme of a second-order accuracy both on spatial and temporal variables Berez (). The theoretical analysis shows that used central-difference scheme is conditionally stable provided that , where and 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. 1) is shown in Fig. 9.
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. 1). As the result, it was obtained a large number of solitons of different amplitudes, their total number for all pulses was 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 of density of number of solitons on amplitudes (see Eq. (9)). The histogram obtained and the model distribution function for the considered illustrative example are shown in Fig. 10. 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.
Iii 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: and 8 cm. Surface waves were generated by a wind blowing up over the water surface with the different mean velocities: 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 V/cm.
Below we present the analysis of only one of the series of experiments with the water depth cm and wind velocity 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 11 presents the Fourier spectrum of wind waves recorded at two distances from the ventilator as indicated above. The critical frequency at cm is s, and to satisfy the condition (29) we cut the Fourier spectra of wind waves at s (see dashed vertical line in Fig. 11) and ignored the high frequency portions of spectra above .
The surface perturbation was reconstructed by means of the inverse Fourier transform on the basis of extracted portions of wave spectra in the range . The fragments of 60-second duration records corresponding to two spatial points of measurements are shown in Fig. 12.
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. 12 were used as the input data for the TKdV equation (1). 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 (1) 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 ; 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 normalised by the total number of all solitons. Figure 13 demonstrates the histogram of soliton numbers versus amplitudes for the time series shown in Fig. 12. The experimental data can be approximated by the Poisson distribution function , where the parameter for the histogram shown in frame (a) and for the histogram shown in frame (b).
The corresponding integral distribution functions for the experimental data of Fig. 12 are shown in Fig. 14 (lines 1) together with the approximative Poisson integral functions with the same parameters as in Fig. 13. The total number of solitons emerged from the wave field shown in Fig. 12a) was 60, and emerged from the wave field shown in Fig. 12b) was 86. As expected, the time series recorded closer to the ventilator (Fig. 12a) contained less number of solitons than the time series recorded further from the ventilator (Fig. 12b). 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. 13a) are uniformly distributed in the time interval of 60 s, then we obtain that the time interval per each soliton is s. As follows from the histogram shown in Fig. 13a) the maximal number of solitons have amplitudes cm and the duration s. Therefore . The similar estimates for the time series shown if Fig. 13b) give the time interval per each soliton s. As follows from the histogram shown in Fig. 13b) the maximal number of solitons in this time series have amplitudes cm and the duration s. Therefore . Thus, we see that in both cases the “soliton gas” is very dense (cf. El-16 (); Shurgalina ()). The fragments of numerical calculations presented in Fig. 15 illustrate the soliton gas density in both time series.
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 Zakh-71 (); Zakh-09 (), 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 El-03 (); ElKam-05 (); Zakh-09 (); Costa (); Dytukh-14 (); Shurgalina (); El-16 (); Carbone-16 (). 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. Shurgalina (); El-16 (). Data processing of laboratory experiments presented in our paper supplement the data processing of field experiments reported in Ref. Costa ().
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 Grimshaw-03 () and Benjamin–Ono Grimshaw-18 () solitons, as well as for the Kadomtsev–Petviashvili lumps Clarke ().
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 (1). The existing numerical codes (see, e.g., Berez ()) 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 (22) 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 II.4.2 shown in Fig. 10, as well as the experimental laboratory data shown in Fig. 13 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. Shurgalina (), 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. Dytukh-14 () 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 advices.
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