# Experimental study of integrable turbulence in shallow water

Ivan Redor, Hervé Michallet, Nicolas Mordant, Eric Barthélemy
<EMAIL_ADDRESS>Laboratoire des Ecoulements Géophysiques et
Industriels, Université Grenoble Alpes, CNRS, Grenoble-INP, F-38000 Grenoble,
France

###### Abstract

We analyze a set of bidirectional wave experiments in a linear wave flume of
which some are conducive to integrable turbulence. In all experiments the
wavemaker forcing is sinusoidal and the wave motion is recorded by seven high-
resolution side-looking cameras. The periodic scattering transform is
implemented and power spectral densities computed to discriminate linear wave
motion states from integrable turbulence and soliton gas. Values of the
wavemaker forcing Ursell number and relative amplitude are required to be
above some threshold values for the integral turbulence to occur. Despite the
unavoidable slow damping, soliton gases achieve stationary states because of
the continuous energy input by the wavemaker. The statistical properties are
given in terms of probability density distribution, skewness and kurtosis. The
route to integrable turbulence, by the disorganization of the wavemaker
induced sinusoidal wave motion, depends on the non-linearity of the waves but
equally on the amplitude amplification and reduction due to the wavemaker
feedback on the wave field.

##  I Introduction

Integrable turbulence is a fascinating topic of nonlinear physics. Integrable
turbulence is theoretically and numerically described in the framework of
integrable equations amongst which the KdV equation [1, 2], the Gardner
equation [3], and the 1D-nonlinear Schrödinger equation [4, 5]. In these
systems an infinite number of degrees of freedom can be excited randomly. As
such there is no energy transfer between these modes and the word turbulence
does not refer to the usual energy cascade between scales. Nevertheless, these
systems can exhibit complex random behaviors that require a statistical
description. Integrable turbulence applies to many fields of physics:
hydrodynamics, optics, and plasmas. [6, 7, 8, 9, 10, 11, 1, 12, 13, 14, 15].

The theory of integrable turbulence in water wave problems is found to be
analytically tractable for two “asymptotic” situations. On the one hand when
the waves are of small amplitude the expansion in powers of non linearity
yields kinetic equations that model wave resonant interactions. In 2D
situations such as for Kadomtsev-Petviashvili type equations [4] resonant
interactions are three-wave resonances. It is known that in the case of the
KdV equation the first nontrivial resonances are five-wave interactions but
with zero amplitudes [16]. This result tends to indicate that integrable weak
wave turbulence in 1D shallow water cases such as for the KdV equations is
precluded.

On the other hand when the turbulence can be considered as a collection of
solitons with random amplitudes and phases, kinetic theories of rarefied
soliton gas [17] or dense ones [18, 19, 20, 21] can be derived . A soliton gas
is thus a random state in which solitons behave as quasi-particles due to the
fact that their collisions are elastic, only altering relative phases, and
thus changing the mean phase speed [17]. Solitons in the shallow water
framework are localized waves which propagate without change of shape due to a
balance between linear dispersive effects that tend to flatten out any surface
perturbations and nonlinear effects that steepen the fronts. Solitons are at
the core of integrable dynamics of the KdV equation.

Empirical confirmation that soliton gases can be generated were given in
optics [22], for deep water waves [23] and for shallow water wave motion [24,
25]. In the experiments energy dissipation cannot be avoided and this seems at
first glance strongly incompatible with the concept of integrability. However,
[24] observed that due to a large scale separation between the nonlinear scale
related to the (short) duration of soliton collisions and the (long)
dissipative time scale, the dynamics is overall ruled by integrability. A
stationary random soliton gas in a long wave flume in shallow water conditions
can thus be sustained with continuous energy input by the wavemaker. Even
though not labeled as soliton gases some 1D flume experiments of [26], in
which the wavemaker has a sinusoidal displacement, lead to random wave
motions. Therefore, the role of the wavemaker in the outcome of these random
stationnary wave states needs to be explored.

$50\,$ years have elapsed between the first theoretical description of soliton
gases by [17] and the first hydrodynamic experiments. A possible reason is the
requirement of highly resolved instruments to capture the space-time evolution
of a random state. Another more fundamental issue relates to the difference
between an infinite or periodic domain setting, usually used in theoretical
approaches, and finite length experimental set-ups. This difference also
combines with how initial conditions are easily prescribed in theory and
numerics while boundary conditions are most of the time the only options at
least in hydrodynamic experiments. A recent notable exception are the deep
water soliton gas experiments by Suret et al. [23] in which the Inverse
Scattering Transform for the 1D Schrödinger equation is used to compute
boundary conditions in a very long flume. In these experiments an ensemble of
random spectral values associated to solitons are prescribed which then evolve
towards a soliton gas.

The constraints on the experiments mentioned above, require to find other
routes to the generation of integrable turbulence and soliton gases, an aspect
investigated in the present work. In this context the question of the
statistical properties of the stationary state of integrable turbulence needs
to be addressed since it remains largely open and was mostly investigated by
numerical studies [1, 2, 27, 28].

In the present study, we first present the experiments on random state wave
motions and the data processing techniques (section II). In section III, we
characterize the conditions for which such random states can be observed by
using the Ursell number and the relative amplitude dimensionless number that
define a phase diagram of our data. A statistical description of the random
states is provided in section IV. Section V is dedicated to the description of
the transition from wave motion to random states.

##  II Experimental setup

The details of the experimental setup and data analysis tools can be found in
[29] and some aspects are discussed in [24, 25] as well. We only recall here
the main features of the experimental set-up.

###  II.1 Wave flume and wave motion measurements

Experiments are performed in the $33.73$ m-long and $55\,$cm-wide LEGI wave
flume with side glass panels. A schematic of the flume is given in Fig. 1 and
a picture in Fig. 2. At one end the waves are generated by a piston-type
wavemaker and opposite a vertical wall ends the flume, in a similar
configuration to that of [30, 26] but in a longer flume. Waves propagate back
and forth in the flume reflecting on the wall and the wavemaker. They are
damped by viscous dissipation in the boundary layers at the bottom and the
sides of the flume. The side glass panels are $1.92$ m long and they are
separated by $8$ cm wide posts that hold them. The water motion is video
recorded through these side glass panels. The cameras have a full HD
resolution ($1920\times 1080$ pixels) with pixel size corresponding to about
$1$ mm in physical space. Each camera records the water contact line motion on
the entire length of the side glass panel. Using $7$ synchronized cameras
running at $20$ frames/s measurements of the water elevation over a $14$
m-long region located at the center of the flume are obtained. The raw images
are rectified by using the image of a calibration grid that was placed in the
flume against the front glass side panel.

Figure 1: Schematics of the wave flume. The flume is $33.73$ m long, $1.3$ m
deep and $55$ cm wide. Waves are generated by a piston-type wavemaker driven
by a computer-controlled hydraulic actuator. The mean piston position is at
$x=0$. $x=9.54\,$m is the right edge of the fifth post. The opposite end of
the flume is a vertical wall on which waves reflect. The side walls are made
of glass with holding posts every $2$m.

In order to improve the contrast of the images for a better accuracy of the
measurements, the bottom of the flume is painted white and the back vertical
panels painted black (see Fig. 3). By choosing adequately the angle of the
camera, the image region just below the contact line appears black while the
region above the contact line is white. This is due to the refraction of the
light rays at the water surface as explained by the schematic in Fig. 3. This
sharp contrast at the contact line allows us to achieve sub-pixel accuracy and
a corresponding resolution better than $0.1$ mm (see [29, 25] for details of
the water contact line measurement). Examples of snapshots recorded by the
cameras are given in Fig. 4 together with the corresponding water free surface
elevation. The set-up provides a time and space resolved measurement of the
wave motion along roughly half the length of the wave flume. An example of a
time-space reconstruction of the water elevation is shown in Fig. 5.

Figure 2: Picture of the flume: the red-brown vertical posts and a USB camera
(bottom right corner).

Figure 3: Schematics of the imaging of the surface elevation. The image
refraction of the white bottom at the water surface creates a sharp contrast
with the image refraction of the black back glass: the water contact line with
the side glass is the boundary between the two images (see sample images in
Fig. 4). Dashed lines correspond to the light rays bounding the image of the
black back glass panel as refracted by the vertical front water boundary with
the glass. The plain lines correspond to the light rays bounding the image of
the white bottom as refracted by the water surface. Figure 4: Sample snapshots
from the $4$ cameras the closest to the wavemaker (top). CAM1 for $9.54\,{\rm
m}\leq x\leq 11.46\,{\rm m}$; CAM2 for $11.54\,{\rm m}\leq x\leq 13.46\,{\rm
m}$; CAM3 for $13.54\,{\rm m}\leq x\leq 15.46\,{\rm m}$; CAM4 for $15.54\,{\rm
m}\leq x\leq 17.46\,{\rm m}$. The images are in inverted gray scale levels and
thus the water surface appears black. Bottom: reconstruction of the water
surface elevation corresponding to the snapshots. The horizontal axis gives
the distance to the wavemaker rest position. H is the measure of the fluid
depth.

###  II.2 Wave generation

Our goal is to obtain integrable turbulence steady states such as soliton
gases. Thus, one would like to generate as many solitons as possible with the
wavemaker. In order to ultimately sustain a large number of solitons in the
flume we take advantage i) of the well-known fission phenomenon in shallow
water of a sine wave train into solitons as evidenced numerically by Zabusky &
Kruskal [31] and observed experimentally by Zabusky & Galvin [32] and in a
more comprehensive way by Trillo et. al [33] and ii) of the amplitude
amplification of the non-linear modes by their interactions with the moving
wavemaker. Mention should be made of the experiments by [30] who, by slightly
detuning the wavemaker motion with respect to the periodic longitudinal
seiching mode of the channel, were able to find a route towards the generation
of integrable turbulence.

Concerning point i), Fig. 5 is an experimental example of the fission of a
sine wave train. Indeed, a sine wave is not a stationary non-linear solution
in shallow-water and integrability imposes an evolution towards a train of
solitons and a non-solitonic background. The space-time representation in Fig.
5 is the full field recorded by the video cameras. The initial $30$ s of the
record is presented corresponding to roughly $3$ wave periods. Leading is the
first sine wave crest that evolves quite differently from the subsequent ones
(bottom panel of Fig. 5). The latter wave crests also undergo steepening due
to nonlinear effects. At $x=10\,$m slight undulations are visible on theses
wave crests corresponding to the regularization (or fission) of the forming
shock by dispersive effects related. After a $23$ m propagation (right of the
field of view) the soliton train is formed with $6$ clearly identified
solitons of decreasing amplitudes. One can also note in top panel of Fig. 5
that the first soliton train which exits the field of view between $10$ and
$17$ s, shows a different number of solitons compared to the subsequent wave
trains. This is due to the fact that the first soliton train propagates on the
rest/initial water level, while the subsequent trains seem to propagate on a
lower (negative) water level since part of initial fluid volume is removed to
generate the train of solitons.

Concerning point ii) the amplification of the non-linear modes by the energy
input provided to waves incoming at the wavemaker will be discussed in detail
in section V.

Figure 5: Fission of a sine wave into solitons. Water depth: $h=12$ cm;
wavemaker sine motion period: $T=10$ s; wavemaker sine motion amplitude:
$a=1.5$ cm. Upper panel: space-time representation of the wave field over the
$14$ m field of view of the cameras and during the first $30$ s. The color-
scale is given in mm with the zero corresponding to the free surface level at
rest. Lower panel: time evolution at $x=18\,$m (dashed line in upper panel).

###  II.3 Spectral and periodic scattering data analysis

####  II.3.1 Fourier power spectrum

Figure 6: Power density spectrum of a soliton gas. Water depth $h=12$ cm;
forcing frequency $f=0.6$ Hz; forcing sine amplitude $a=1.5$ cm; Ursell number
$U=0.53$. Top panel: 2D Fourier transform. Bottom panel: 1D frequency Fourier
transform; blue plain line: total signal ($\eta_{t}$); red dashed line: right-
running waves ($\eta_{r}$); yellow bold line: left-running waves ($\eta_{l}$).

A standard tool to study wave propagation is the 2D Fourier power spectral
density [34] defined here as:

$E(k,\omega)=\frac{1}{2\pi
L\,T}\,\Bigg{\langle}\,\left|\iint\eta(x,t)\,e^{i(kx+\omega t)}\,{\rm
d}t\,{\rm d}x\right|^{2}\Bigg{\rangle}$ (1)

where $\eta(x,t)$ is the free surface displacement. The space integral for $x$
covers the field of view of length $L$ of the cameras and the time $t$
integral spans a time window of duration $T$. It is notable that the Fourier
transforms are actually discrete Fourier transforms due to discrete sampling
in space and time. The statistical average represented by $\langle.\rangle$,
is an average over successive temporal windows using the standard Welch
method. An example of such a spectrum is shown in Fig. 6(a). Of note several
ridges of energy higher than that of the background with the following
interpretations:

  * •

ridge a: signature of weak dispersive shallow water waves following the Airy
dispersion relation

$\omega^{2}=g\,k\,\tanh(k\,h).$ (2)

These dispersive waves originate from the bound waves of the wavemaker
monochromatic sinusoidal wave forcing and from the weak non integrable effects
during soliton collisions [29, 24];

  * •

ridge b: signature of shallow water non-linear waves which Fourier modes all
travel at a velocity close to $c_{0}=\sqrt{g\,h}$.

  * •

ridges c: signature of transverse waves. The uni-nodal transverse wave would
be at a frequency of $0.9\,$Hz. The energy of the uni-nodal transverse waves
is three orders of magnitude smaller than that of longitudinal waves.

Waves corresponding to the upper-right quadrant of the Fourier space ($k<0$
and $\omega>0$) propagate to the right and those corresponding to the lower-
right quadrant ($k>0$ and $\omega>0$) propagate to the left. Thus, by
selecting a specific quadrant $(k,\omega)$ of the Fourier transform and
inverse transforming back to the physical space, it is possible to separate
waves going into opposite directions (see an example in Fig. 7). An
alternative for this separation is the Radon transform as discussed in [25].
We also use the time 1D frequency Fourier power spectrum density for the
analysis of the signals. An example of such a spectrum is shown in Fig. 6(b)
either for the full signal, for the split into right- and left-propagating
waves.

Figure 7: Splitting of the bidirectional wave field into unidirectional
fields, case of a soliton gas. Top panel: the total wave motion field
$\eta_{t}$; middle panel: the right-running waves $\eta_{r}$; bottom panel:
the left-running waves $\eta_{l}$. Water depth $h=12$ cm; wavemaker sine
motion period: $T=10$ s; wavemaker sine motion amplitude: $a=1.5$ cm. Color
scale in centimeters.

The case described in Fig. 6 and 7 and analyzed in [24] is typically an
example of integrable turbulence containing a significant number of solitons.
Solitons are responsible for “ridge b” of Fig. 6 and they also clearly leave
straight line signatures in the $(x,t)$ plane of Fig. 7. This was discussed in
detail in [25].

####  II.3.2 Periodic Scattering Transform

While the Fourier Transform (FT) is the adequate processing tool to study the
propagation of linear waves, it is unfit to discriminate an ensemble of
nonlinear coherent waves such as solitons. Because of the nonlinear
interactions, the FT spectral components of evolving nonlinear waves are not
invariants as for linear waves.

Gardner et al. [35] made a significant leap forward with the Direct Scattering
Transform (DST) of the KdV equation that extracts the spectrum of the
associated Schrödinger equation which potential is the nonlinear wave signal
to be analyzed. This method decomposes a time series into nonlinear solitonic
modes that evolve independently in time and space without change of shape
along with radiating shallow water weakly nonlinear waves. Once the time
independent spectrum of nonlinear modes is determined, the signal can be
reconstructed at any time by the Inverse Scattering Transform (IST). DST
theory for the case of a localized initial condition [35] differs from the
much more complex case of the spatially periodic initial condition known as
the finite band theory [36, 37, 38, 39]. In our experimental case we are more
concerned with the periodic case due to our configuration in which the waves
propagate back and forth in the flume. Although the waves in the present
experiments are not in most cases periodic, the wave motion is however
confined in space and does not decay at infinity. This approach has been used
by [40] to analyze oceanic field data.

We implemented the Periodic Scattering Transform algorithm (PST) for the KdV
equation developed by Osborne [41, 42, 43] in a program to compute the
nonlinear spectrum of the experimental free surface records. In the periodic
case, the non linear modes are described by hyper-elliptic functions. These
waves similar to cnoidal waves are characterized by the so-called spectral
modulus $m$ which quantifies the level of nonlinearity and which is an output
of the PST [42]. For a vanishing small modulus the modes are close to sine
waves. For $m$ very close to $1$, the modes appear as localized pulses in the
periodic box that resemble solitons. The delineation between solitonic modes
and radiation modes thus breaks down to the choice of the threshold modulus.
This threshold modulus also defines the reference depth $h_{ref}$ [41, 42, 43]
on which these solitonic modes propagate. Osborne [41] suggests that modes
with $m>0.99$ can be considered as solitons a definition also used by [33]. We
will not discuss here the details of our implementation of the PST which is
described in [29] with different validation cases and an analysis of various
limitations since this method has been validated various times [39, 33]. The
main output of the PST is the spectrum that lists the nonlinear modes and
their moduli.

It must be noted that the present implementation of the PST is related to the
KdV equation framework which describes unidirectional wave propagation only.
Thus, for a consistent application of the PST to our measurements, we first
separate left and right-propagating waves using the $(k,\omega)$ Fourier
spectrum as described above and then apply the PST either to left or right-
propagating waves.

Among the issues related to PST and discussed in [29] one can mention the case
of a signal containing solitonic modes of equal or very close amplitudes for
which the detection of the modes is challenging since the eigenvalues of the
spectrum are very close. Another issue raised in [29] is the impact of the
duration of the signal sample on the number of solitons and the reference
depth especially for a soliton gas. Indeed, the longer the signal the more
solitonic modes will be detected and therefore the lower the reference level.
A last issue is related to the arbitrary threshold and the impact of solitons
which moduli are very close to the threshold modulus. This issue is
illustrated in Fig. 8 related to the fission of the sine into solitons as
already discussed above.

Figure 8: PST analysis of the fission of a sine wave ($U=19.4$) into solitons.
Water depth $h=12$ cm. Sine motion forcing period $T=10$ s and amplitude
$a=1.5$ cm. Top panel a): free surface elevation $\eta$ at different positions
along the flume, black: at the wavemaker $x=0$; blue: $x=10$ m; red: $x=23$ m;
The dashed lines are the reference levels at each $x$ locations as computed by
the PST with the choice of $m=0.99$ threshold modulus. Bottom panel b): PST
modal amplitudes $a_{n}$ ($\bigtriangledown$) and module $m$ ($+$) of the
different modes of number $n$ at different position along the flume (same
color coding as top panel). Full symbols: threshold modulus at $m=0.99$; empty
symbols: threshold modulus at $m=0.985$.

The PST has been applied to the three measured wave elevation profiles shown
in Fig. 8(a). The corresponding moduli and wave amplitudes (triangles with
same colors) as computed with the PST are plotted in Fig. 8(b). A threshold
modulus of $0.99$ yields $9$ solitonic modes at $x=0$ and $x=10$ m but only
$8$ solitons for the signal at $x=23$ m. Indeed, the modulus of the $9$th mode
at $x=23$ m drops slightly below the threshold modulus $0.99$. Would the
threshold be lowered down to $0.985$, the $9$th mode at $x=23$ m would be
considered as a soliton. The soliton amplitudes are seen to be quite close in
the three cases with a small decay with distance due to dissipation. Moreover
this reduction of amplitude with $x$ is enhanced by the fact that since the
$9$th mode is no longer a soliton at $x=23$ m the reference level
automatically raises reducing the soliton amplitudes. The amplitudes with
threshold at $0.985$, at $x=23$ m, are shown in Fig. 8(b) as empty triangles
and are seen to be larger than the ones for the $0.99$ threshold (filled
triangles) and significantly closer to the ones obtained at $0$ m and $10$ m
except for a slight decay due to dissipation. Although both sets of amplitude
are globally consistent with the measurement, the empty symbols are closer to
the measurements, at least for the large solitons. This illustrates the effect
of the choice of a threshold modulus in the case of a large number of
nonlinear modes which amplitudes and moduli have a wide distribution.

##  III Parametric study of the stationary regime

In the previous sections we discussed the transient evolution of a sine wave
and recalled that some conditions produce an integrable turbulence. However,
is it the fate of all wave conditions to evolve into a random wave motion
state? Two examples of wave motion recordings are plotted in Fig. 9
corresponding to two different regimes. These recordings are taken well after
the start of the experiment, at roughly $16\,$mn. As time goes, one remains
periodic while the second becomes disorganized or random. What makes these $2$
cases different?

Figure 9: Time series of the free surface displacement forced by a sine wave
prescribed at the wavemaker. Water depth is $h=12\,$cm; top panel: wavemaker
frequency $f=0.25\,$Hz, amplitude $a=0.25\;$cm, Ursell number $U=0.52$; bottom
panel: wavemaker frequency $f=0.6\,$Hz, amplitude $a=1.5\;$cm, Ursell number
$U=0.53$ (these cases are also in Fig. 10).

The sine wave forcing at the wavemaker depends on $3$ physical parameters: the
amplitude $a$ of the sinusoidal wave, the period $T$ (or the wavenumber
$k=2\pi/(c_{0}\,T)$ and the water depth $h$. The space of parameters was
explored by changing the values of these $3$ parameters. The amplitude was
typically varied in a range $a\in[0.125,1.5]$ cm, the period in a range
$T\in[1.6,5.5]$ s (or the frequency $f\in[0.18,0.6]$ Hz) and the water depth
was set to the values $10$, $12$, $16$ and $20$ cm.

One of the relevant non-dimensional number in the shallow water context is the
Ursell number [44]. It is even the only dimensionless number for KdV
unidirectional wave motion dynamics. The Ursell number is the ratio in order
of magnitude of the nonlinear to dispersive terms of the KdV equation. The
dimensional version of the KdV equation is,

$\partial_{t}\eta+c_{0}\,\partial_{x}\eta+\frac{3\,c_{0}}{2\,h}\,\eta\,\partial_{x}\eta+\frac{h^{2}\,c_{0}}{6}\,\partial_{xxx}\eta=0.$
(3)

Consider the following scaled dependent and independent variables [27],

$\eta\leftarrow\frac{3}{2}\,\frac{h^{2}}{l^{2}a}\,\eta\ \ \ \ \ \
x\leftarrow\frac{\sqrt{6}}{h}\,(x-c_{0}\,t)\ \ \ \ \ \
t\leftarrow\frac{\sqrt{6}}{h}\,c_{0}\,t$ (4)

where $h$ is the water depth at rest, $a$ a characteristic vertical length
scale (the amplitude of the forcing wave for instance) and $l$ a horizontal
length scale (the forcing wave wave length). Then equation (3) scales to,

$\displaystyle\partial_{t}\eta+R\,\eta\,\partial_{x}\eta+\partial_{xxx}\eta=0$
(5) $\displaystyle R=\frac{a\,l^{2}}{h^{3}}$ (6)

where $R$ is a Ursell number. Different pre-factors can be appended to $R$.
Hereafter we use the following definition of the Ursell number used by [45],

$U=\frac{3}{4}\,\frac{a}{k^{2}\,h^{3}}$ (7)

where $k=2\,\pi/l$ is the wave number. This writing is exactly the ratio of
the amplitudes of the second-order to the first-order term in the Stokes
expansion of the water wave problem. For a time series the Ursell number would
be,

$U=\frac{3}{16\pi^{2}}\,\frac{a\,c_{0}^{2}\,T^{2}}{h^{3}}$ (8)

Equation (5) shows that for very small $U$ the equation becomes linear and
dispersive, the so-called Airy equation. Under this condition the forcing wave
remains linear but disperses (different frequency components propagate at
different speeds) as it propagates and no soliton emerges. For large values of
$U$ equation (5) becomes non-linear of the Burgers type. A sinusoidal forcing
wave that fulfills such condition will undergo nonlinear steepening up to the
gradient catastrophe producing a steep front face (shock wave). At that point
the wave front face characteristic length is small and dispersion comes into
play. Dispersion forces the shock wave to fission into a train of solitons
[31, 46, 33]. The Ursell number of the forcing wave therefore indicates how
many solitons are expected [33] . Since our set-up does not allow for initial
condition recurrence to take place because of the end wall reflection, the
Ursell number also measures how disorganized the regime is. Indeed, solitons
will reflect back on the end wall and the wavemaker, interacting with others,
generating phase shifts and therefore possibly disorganizing the initial
periodicity.

Figure 10: Power density spectra of the surface elevation for experiments with
the same Ursell number, $U\simeq 0.53$, and differing values of the forcing
amplitude $a$ and frequency $f$. Left-running waves at $x=40\,$m. Depth,
$h=12\,$cm. Vertical dashed line indicates the wavemaker forcing frequency.
From top to bottom and left to right: $I_{m}=2.54$, $I_{m}=3.93$,
$I_{m}=3.38$, $I_{m}=1.12$, $I_{m}=0.40$, $I_{m}=0.19$, $I_{m}=0.16$,
$I_{m}=0.20$, $I_{m}=0.25$ (see (9) in text for the definition of $I_{m}$).

However, we observed that the sole value of the forcing wave Ursell number $U$
is insufficient to discriminate between the periodic and random states. Fig.
10 shows the frequency spectra of $9$ experiments for the left-running
component of the wave motion at $x=40\,$m which is the best compromise between
soliton separation after fission and dissipation. All these experiments have
very close Ursell number values of $U=0.53$, but distinct values of the
frequency $f$ and amplitude $a$. Signatures of different stationary states are
observed from very periodic ones at low forcing frequency and low forcing
amplitude to random ones at high forcing frequency and high forcing amplitude
and going through continuously varying power spectra shape in between. This
observation shows that another dimensionless parameter must be taken into
account to sort the different states out. Fig. 11 shows the corresponding
distribution of non-linear mode amplitudes given by the PST. The periodic
cases (3 top subplots of Fig. 11 and correspondingly in Fig. 10) exhibit sets
of non-linear modes of nearly constant amplitudes indicating that these cases
remain strongly organized. For instance the case $f=0.18\,$Hz for a $60$ s
wave motion recording corresponds to roughly $10$ periods of the forcing wave
for which the PST gives $10$ soliton modes of $0.15\,$cm amplitudes and hardly
any other modes. A small soliton of $0.15\,$cm amplitude is locked to each
wave period. By contrast the soliton modes in the random cases are much more
numerous and their amplitudes distributed over a large range indicating indeed
that these cases are random. These states (3 bottom subplots of Fig. 11 and
correspondingly in Figure 10) are considered to be what is called integrable
turbulence [24]. Discussion of the amplitude distribution is post-poned to
section IV.

Figure 11: Solitonic mode amplitude $a_{s}$ distribution versus mode number
$n$ computed by PST; same data as in figure 10: $U\simeq 0.53$, left-running
waves at $x=40\,$m, $h=12\,$cm.

To delineate integrable turbulence from other stationary states, the forcing
wave non-linearity ratio $a/h$ is necessary where $a$ is the wavemaker
amplitude. Fig. 12 shows that states with identical $U$ and $a/h$ values but
with differing $h$ have power spectra of similar shape either continuous,
peaked or mixed. It therefore appears that the forcing wave dimensionless
numbers $U$ and $a/h$ are the main dimensionless numbers to discriminate
between various stationary wave motion states.

Figure 12: Examples of PSD of experiments with the same Ursell number $U$ and
same values of $a/h$ and different water depth $h$. left panel: $U=0.31$ &
$a/h=0.105$; middle panel: $U=0.52$ & $a/h=0.0625$; right panel: $U=0.28$ &
$a/h=0.021$.

Figure 13: Phase plane of the flume experiments. Forcing wave non-
dimensionless parameters: $U$ and $a/h$. The size of the circles relate to the
water depth $h$ and the color scale to the value of the randomness index
$I_{m}$ defined by (9). Values of $I_{m}<0.9$ correspond to soliton gaz
regimes.

We conclude from these observations that another way to synthetically
discriminate the various regimes, within our experimental framework, is to
compute the frequency spectrum as given in Fig. 10. In the periodic case, the
spectrum is mostly made of narrow peaks with the fundamental peak at the
forcing frequency and the other peaks corresponding to higher harmonics. In
the random case the spectrum is quite distinct and it is seen to be continuous
with a flat plateau [1, 24] that extends from the forcing frequency down to
the smallest resolved frequency. At frequencies higher than the forcing
frequency the spectrum decays exponentially. At this point, a question arises
on how to quantify the state of the wave field. At this end, we define a
dimensionless randomness index $I_{m}$ based on the shape of the power
spectrum:

$I_{m}=\log_{10}\left(\frac{E_{forcing}}{E_{av}}\right)$ (9)

where $E_{forcing}$ is the spectrum value at the forcing frequency (for a
given value of the duration of the signal, chosen at 60 s here) and $E_{av}$
is the average value of the spectrum on the range of frequencies lower than
that of the forcing. For a periodic signal this index is large since the
spectrum is strongly peaked at the forcing frequency. In the random case, no
peak is present at the forcing frequency and the spectrum at lower frequencies
is flat containing most of the energy making the index very small.

The phase diagram in Fig. 13 compiles the various experiments we conducted.
The size of the circles indicates what the water depth $h$ is and the color
filling the value of the index $I_{m}$. Blue colors below $1$ can be
considered as random states (integrable turbulence) and red/brownish colors
are organized states with a certain degree of periodicity. We need to
emphasize that we restricted ourselves to relatively low levels of
nonlinearity to prevent wave breaking both at the wave generation but also by
wave interaction within the flume. At the highest values of $a/h$ micro-
overturning may still have happened occasionally for very large magnitude of
the water elevation due to superimposition of many solitons, but without
impacting significantly the global state. In this phase diagram a wedge
emerges separating organized states from integrable turbulence. The wedge
containing random states is for $a/h$ greater than roughly $0.07$ and $U$
larger than roughly $0.2$. The boundaries of the wedge come with
uncertainties. A more accurate identification would require other lengthy
measurements since each point in the phase plane corresponds to an experiment
duration at least 1 hour-long. Furthermore, note that close to the wedge
boundaries, experiments with very close $a/h$ and $U$ values but with distinct
$I_{m}$ (distinct colors) are nearly superimposed suggesting subdominant
dependencies to other physical parameters. One obvious extra parameter is the
length of the flume (see the discussion in section V). The points at the far
right with $U\simeq 19.3$ that lie in the wedge should lead to fission in
soliton trains contradicting an index $I_{m}$ value above $1$. In these
experiments the wavelength of the forcing wave is roughly $10\,$m. Each wave
length is too long to completely fission into solitons before the end wall
reflection: solitons never really separate before they are dissipated by
friction. At the other end of the phase diagram experiments with $a/h=0.08$
and $h=20\,$cm are situations of intermediate water depth with $k\,h>0.314$
not prone to soliton generation and therefore to the generation of integrable
turbulence.

##  IV Statistics of the stationary regime

In this section statistical distributions and moments of the experimental
integrable turbulence are presented. Statistical information complements PST
analysis. PST in our present study is a key tool to assess the existence of a
soliton gas and characterize the modal content of the gas. Nonetheless
integrable turbulence encompasses also shallow water situations where
solitonic modes coexist with radiation modes. As discussed in the introduction
the literature on the statistical description of integrable turbulence is
sparse while such situation can be present in field measurements [40]. A
noticeable exception is the numerical study by [1] on KdV random wave fields,
a reference study on these aspects.

In Fig. 14 we plot the free surface displacement empirical probability density
distribution for $U=0.53$ (case of Figs. 6 and 7 and bottom-right-hand
subplots in Figs. 10 and 11). On the same plot the Gaussian distribution is
the narrow-band linear sea state theoretical distribution [47]. Our $U=0.53$
is very close to the Ursell $Ur=0.95$ of [1] (different definition) and will
serve for comparison. In Fig. 14, the three different probability
distributions correspond to the total time series, the right-running and left-
running waves. Clearly each of these distributions is positively skewed
(higher area under the distribution to the right of the mode). The positive
skewness is a well known feature of non-linear waves with peaky crests and
wide shallow troughs. The left-running wave elevation distribution tends to be
more symmetric as expected due to dissipation. Indeed, the left-running wave
are those reflected at the end wall travelling a longer distance from the
wavemaker compared to right-running waves. By the Gram-Charlier expansion [47]
including the skewness, the Gaussian distribution is corrected and fits the
experimental data fairly well.

Figure 14: Free surface elevation $\eta$ probability density function computed
over $175$ points (every $8\,$cm from $x=9.53$ m up to $x=23.45$ m) and for
$25\,$mn in the stationary regime. The thin blue line is the Gaussian
distribution. The thick red dashed line is the Gaussian distribution corrected
with the skewness of the total wave record. Case $U=0.53$, $f=0.6\,$Hz,
$a=1.5\,$cm, standard deviation $\sigma_{t}=1.2\,$cm, $\sigma_{r}=0.98\,$cm,
$\sigma_{l}=0.69\,$cm for resp. the total time series ($\circ$), the right-
running part of the time series ($\rhd$) and the left-running part of the time
series ($\lhd$).

For a narrow-band Gaussian linear sea state, the Rayleigh distribution rules
the distribution of crest levels [47]. Crest level distributions are also used
by [1] to characterize their random nonlinear shallow water wave field
obtained numerically in the KdV framework. The Rayleigh one parameter
probability density function $p(A)$ and exceedance probability distribution
$P(A)$ write,

$\displaystyle P(A)$ $\displaystyle=$ $\displaystyle
e^{\displaystyle{-2A^{2}}}$ (10) $\displaystyle p(A)$ $\displaystyle=$
$\displaystyle 4\,A\,e^{\displaystyle{-2A^{2}}}$ (11) $\displaystyle A$
$\displaystyle=$ $\displaystyle\frac{A_{c}}{A_{s}}$ (12) $\displaystyle A_{s}$
$\displaystyle=$ $\displaystyle 2\,\sigma$ (13)

where $A_{c}$ are the crests levels determined by a zero-crossing procedure
[48], $\sigma$ the standard deviation of the free surface displacement [1].
Both the experimental and linear wave theoretical exceedance probability are
plotted in Fig. 15 for the $U=0.53$ case of soliton gas. As already noticed by
[1] the experimental crest exceedance probability lies above the theoretical
Rayleigh probability function. The larger crests tend to be more frequent in
an integrable turbulence than in a random field of waves complying to the
Rayleigh distribution. Among the larger crests are those of course of the
solitons that populate this integrable turbulence. The present $U=0.53$ close
to the $Ur=0.95$ of [1] shows that the experimental exceedance probability
tallies quite nicely with the numerical one (see figure 10 of [1]). The
correction to the Rayleigh distribution suggested by [49] (their equations
(5.19) et (5.20)), which takes into account the free surface displacement
skewness, is also plotted. While this correction matches the experimental
values for small $A$, it overestimates the Rayleigh distribution for large
amplitudes corresponding to the large solitons. It would suggest that higher
order moments are important.

Figure 15: Crest elevation $A$ exceedance probability distribution. Thin blue
line: Rayleigh distribution. Thick red dashed line: skewness corrected
Rayleigh distribution (10). Case $U=0.53$, $f=0.6\,$Hz, $a=1.5\,$cm, standard
deviation $\sigma_{t}=1.2\,$cm, $\sigma_{r}=0.98\,$cm, $\sigma_{l}=0.69\,$cm
for resp. the total time series ($\circ$), the right-running part of the time
series ($\rhd$) and the left-running part of the time series ($\lhd$).

The moments that characterize best the empirical distribution in Fig. 14 are
the skewness ${S}$ and the kurtosis ${K}$, defined for $N$ values equally
spaced in time by,

$\displaystyle{S}=\frac{\mu_{3}}{\sigma^{3}}$ (14)
$\displaystyle{K}=\frac{\mu_{4}}{\sigma^{4}}$ (15)
$\displaystyle\sigma^{2}=\frac{1}{N}\,\sum_{i}\left(\eta_{i}-\overline{\eta}\right)^{2};$
(16)
$\displaystyle\mu_{3}=\frac{1}{N}\,\sum_{i}\left(\eta_{i}-\overline{\eta}\right)^{3};\
\ \ \ \mu_{4}=\frac{1}{N}\,\sum_{i}\left(\eta_{i}-\overline{\eta}\right)^{4}$
(17)

Where $\overline{(.)}$ stands for the time average. For a Gaussian random wave
field ${S}=0$ and ${K}=3$. The kurtosis measures the heaviness of the
probability distribution tails while the skewness indicates the asymmetry of
the distribution around the average. The experimental values for the skewness
and the kurtosis for the left-running waves in the $U=0.53$ case are
${S}=0.70$ and ${K}=3.40$. The corresponding values of [1] are ${S}=0.73$ and
${K}=3.45$. Our experiments and their numerical simulations yield very close
values. We compared their numerics with the experimental left-running wave
field because the latter is deemed to have achieved statistical stationarity.
While our experimental procedure starting from a sine wave differs from the
initial conditions of [1], both generate a very similar long term integrable
turbulence stationary state which is mainly characterized by the Ursell
number.

In their numerical study, [1] show that for $U>0.16$, both the skewness and
the kurtosis increase quasi-linearly with the Ursell number (see their figure
7) suggesting that the skewness and the kurtosis are linearly related. We show
in Fig. 16 that this also stands in our experiments. In this figure all the
experimental runs labeled as integrable turbulence, that is with a mixing
index $I_{m}<1$, are plotted and clearly align.

Figure 16: Skewness ${S}$ of the free surface elevation versus the kurtosis
${K}$. Colors refer to the level of the mixing index $I_{m}$. The dashed line
is the lower bound of the Kurtosis: ${K}\geq 1+{S}^{2}$ ([50] p.432). Data
corresponding to points in Fig. 13 with $I_{m}<0.9$.

Figure 17: Crest elevation $A$ probability density function. Thin blue line:
Rayleigh pdf (11). Thick red dashed line: skewness corrected Rayleigh pdf
[49]. Case $U=0.53$, $f=0.6\,$Hz, $a=1.5\,$cm, standard deviation
$\sigma_{t}=1.2\,$cm, $\sigma_{r}=0.98\,$cm, $\sigma_{l}=0.69\,$cm for resp.
the total time series ($\circ$), the right-running part of the time series
($\rhd$) and the left-running part of the time series ($\lhd$).

The pdf of the crests levels is plotted in Fig. 17. At small amplitude this
pdf shows that the empirical distribution exhibits a gap instead of a maximum
around the crest value of $0.5$. The data departs from the Rayleigh pdf and
this discrepancy is badly explained with the skewness modified Rayleigh pdf.
It appears that crests levels follow a kind of bimodal distribution. Some of
the crests detected are probably those of solitons but not all of them.
Consequently the latter statistical characterizations do not inform on how
close a given state of integrable turbulence is to a soliton gas. In contrast
the PST provides quantitative information on the solitonic content of the
times series. Of interest are the statistics of the amplitudes of the
solitonic modes that relate to the finite-band spectrum of the Schrödinger
equation in the PST. In Fig. 18 the empirical amplitude probability histograms
are plotted at $4$ different locations along the flume for the Ursell number
$U=0.53$ case. The PST is run on $55$ overlapping time segments of $60\,$s for
right-running waves at $x=10.3\,$m and $x=22.78\,$m and for left-running ones
at $x=44.68\,$m and $x=57.16\,$m. The time window of $60\,$s is slightly
smaller than the time necessary for a wave to travel twice a total flume
length ($62.2\,$s for a depth of $12\,$cm ) which includes a reflection on the
end wall and one on the wavemaker. This ensures we process a time series
excluding waves measured twice.

The empirical histograms all exhibit $2$ wide peaks of solitonic modes as if
the distributions were the combination of two distributions that characterize
the experimental integrable turbulence, an aspect pointed out in Fig. 17. As
the wave trains travel back and forth, both the width of the histograms and
the largest amplitude peak of the distribution decrease. In the ideal KdV
integrable turbulence these distributions should be invariant with $x$. In the
experiments as the solitons propagate, dissipation progressively reduces their
amplitude according to the Keulegan dissipation “law” [51]. The Keulegan
dissipation law reads,

$\displaystyle\left(\frac{a}{h}\right)^{-1/4}=\left(\frac{a_{0}}{h}\right)^{-1/4}+\,D\,\frac{x}{h}$
(18) $\displaystyle
D=\frac{1}{12}\,\left(1+\frac{2\,h}{w}\right)\,\sqrt{\frac{\nu}{g^{1/2}\,h^{3/2}}}$
(19)

where $w$ is the width of the flume, $\nu$ the water viscosity and $a_{0}$ the
initial soliton amplitude. This Keulegan law has been thoroughly validated
experimentally [51, 52, 25]. In the dissipation process the large solitons
become intermediate amplitude solitons that tend to populate the initial gap
around $1.5\,$cm. Moreover as the largest decrease more rapidly than the
smallest, as given by the Keulegan dissipation “law” (18), the distribution is
severely eroded from the right end of large amplitudes. In Fig. 18 the initial
distribution of right-running waves at $x=10.3\,$m is transformed into the
distribution at the next location ($x=22.78\,$m) by applying Keulegan
amplitude reduction law (18). The initial distribution is likewise
“transformed” to the distributions at other locations up to and back to
$x=10.3\,$m. It is assumed that only reflections take place at the end wall
and at the wavemaker. The successive transformed distributions compare well
with the measured ones except at $x=10.3\,$m. At this location the measured
wave distribution is much wider with peak values at higher amplitudes than
that of the Keulegan-transformed distribution. This indicates that the wave
train gains energy as it bounces back on the wavemaker. On one hand it is due
to interactions of the various waves incoming on the moving wavemaker and on
the other hand to the generation of new waves by the wavemaker. By positive or
negative feedback some waves have an amplitude that increases and the contrary
for others. The net result is an increase in the wave energy flux at
$x=10.3\,$m. This is confirmed by PST analyzing this $U=0.53$ sinusoidal
wavemaker forcing wave. The PST yields one set of $n$ modes of modulus larger
than $0.99$, thus in practice a set of $n$ amplitudes very close to
$2.84\,$cm. Nonlinear modes of lower modulus are also detected in this PST
processing. Thus, the modal content of the forcing sinusoid is one solitonic
mode and at least one cnoidal type mode per period. The solitonic mode
amplitude is smaller than that of the highest amplitude peak at $x=10.3\,$m
which is around $3.5\,$cm. The likely explanation is that in the stationary
regime the nonlinear modes of the forcing wave interact positively with the
moving wavemaker to a point where the solitonic mode amplitude increases and
the most nonlinear cnoidal mode becomes a solitonic mode of modulus $m>0.99$.
The feedback of the wavemaker on the travelling waves produces $2$ sets of
solitonic modes.

The mechanism by which soliton content and amplitudes change, are now
discussed in more detail by analyzing the transient route to integrable
turbulence.

Figure 18: PST soliton amplitudes histograms at different locations in the
flume. Top panels correspond to right-running waves. Bottom panels correspond
to left-running waves. Histograms with bars: PST processed experimental data;
dashed lines predicted histograms by applying the Keulegan damping law (18);
green dashed: propagated histogram from $x=57.16\,$m to $x=10.3\,$m as if the
wavemaker was a fixed vertical boundary. Case $U=0.53$, $f=0.6\,$Hz,
$a=1.5\,$cm and $\nu=1.2\,10^{-6}\,{\rm m^{2}/s}$

##  V Transition to integrable turbulence: randomization of the periodic
forcing

Figure 19: Attenuation and amplification of right-running wave trains. Initial
free surface elevation $\eta$ at $x=10.5$ m. Top panel: $a=1.5$ cm, $f=0.6$ Hz
($T=1.67$ s), water depth $h=16\,$cm, $a/h=0.094$ and $U=0.30$. bottom panel:
$a=0.75$ cm, $f=0.43$ Hz ($T=2.33$ s), water depth $h=12\,$cm, $a/h=0.062$ and
$U=0.53$. Vertical dashed lines separated by the time necessary for a wave to
acheive a round trip of $2\times 33.73\,{\rm m}$ at $c_{0}$.

In a periodic box and with an initial condition made of a sine wave, Zabuski &
Kruskal [31] numerically predict a recurrence, i.e. the fact that the wave
system retrieves in a finite time a state very close to the initial condition
[31]. In our experimental setup, it is not possible to start from an arbitrary
initial condition which is not rest. What can only be achieved in a controlled
way, is to start wave forcing at one end with a wavemaker and a body of water
at rest in the entire flume.

The reflection on the moving wavemaker is a key point to the transition
towards integrable turbulence. Indeed, a wave that travels from and back to
the wavemaker, after reflection on the end fixed wall, will interfere with the
wavemaker motion. Depending on the relative phase of the wave with respect to
that of the wavemaker, the wave amplitude may increase or decrease. A well
known situation is that of a standing wave. In Fig. 19 two experimental
examples of such positive or negative interference are given. The wave trains
plotted in Fig. 19 are right-running filtered waves. To assess if an
amplification or attenuation takes place the one way filtering is necessary to
suppress any partial standing wave pattern, with partial nodes and anti-nodes,
which blur the right running wave amplitudes. The bottom panel is that of an
amplification for $h=12\,$cm ($c_{0}=1.08\,{\rm m/s}$). The wave crest marked
with a dashed vertical line at $t=128.25$ s travels away and a complete round
trip to be recorded as a right running wave at $x=10.5$ m at $t=190.7$ s. This
crest is almost in phase with a newly generated wave producing a positive
interference.

By contrast in the top panel, the right-running waves undergo an attenuation
for $h=16\,$cm ($c_{0}=1.25\,{\rm m/s}$). The wave crest marked with a dashed
vertical line at $t=225.25$ s travels away and a complete round trip to be
recorded as a right running wave at $x=10.5$ m at $t=279.1$ s. The wave crest
after a complete round trip is out of phase with a newly created wave (in
phase with the trough) producing a negative interference and therefore an
attenuation. In section III the length of the flume was characterized as a
subdominant parameter. Indeed, a slight change of length or equivalently of
wavemaker frequency can possibly shift the feedback of the wavemaker with the
traveling waves from positive to negative and conversely.

Figure 20: Transition to integrable turbulence. Free surface elevations $\eta$
at $x=10.5$ m for 3 different wave amplitude forcing. Wave-maker frequency
$f=0.3$ Hz and water depth at rest $h=12$ cm. Top panel (a) $a=0.5$ cm,
$a/h=0.04$ and $U=0.72$; middle panel (b): $a=1.0$ cm, $a/h=0.08$ and
$U=1.44$, vertical dashed lines separated by the time necessary for a wave to
complete a round trip in the flume; bottom panel (c): $a=1.5$ cm, $a/h=0.12$
and $U=2.16$.

Fig. 20 shows water elevation records at $10.5$ m from the wavemaker with
three conditions of forcing with same frequency and water depth but distinct
amplitudes and thus distinct values of nonlinearity level $a/h$ and Ursell
number $U$.

In the first case (a), for the lowest value of $a/h$, wave crest elevations
evolve by discrete set of steps (either increasing or decreasing) every round
trip (every $62.2$ s), but on the long run the signal becomes periodic and the
crest levels are smaller than initially. The soliton content processed by PST
gives $1$ soliton of amplitude $a_{s}=0.76\,$cm per period in the initial
first round trip (either with threshold $m=0.99$ or $m=0.999$) and no solitons
at all in the periodic tail of the signal. The negative feedback produced by
the wavemaker inhibits the survival of solitons.

In the intermediate case (b) even though at each round trip the waves overall
are slightly attenuating for $t<200\,$s there are small time intervals of the
signal that show irregular crest level increase. These time intervals occur
roughly every round trip of $62.2\,$s, indicated by the vertical dashed red
lines and widen at each round trip. These amplification sequences eventually
overlap after $t=700\,$s to produce a random state. The PST analysis, at the
threshold $m=0.99$, of $6$ periods in the first round trip gives $12$ solitons
that is $2$ per period with average amplitudes of respectively
$a_{s1}=2.25\,$cm and $a_{s2}=1.76\,$cm. The equivalent analysis on sequences
of $6\times T$ duration beyond $t=700\,$s yields an average of $16$ solitons
with the largest amplitude at $a_{s}=3.9\,$cm.

In case (c), the disorganization is even faster than in case (b). The PST
analysis, at the threshold $m=0.99$, of $6$ periods in the first round trip
gives $3$ solitons per period with amplitudes of respectively
$a_{s1}=3.95\,$cm $a_{s2}=2.43\,$cm and $a_{s3}=0.92\,$cm. The equivalent
analysis on sequences of $6\times T$ duration beyond $t=2750\,$s yields an
average of $21$ solitons with the largest amplitude at $a_{s}=6.1\,$cm.

The former last two cases clearly indicate on the one hand that even though
the initial wave trains contain solitons the stationnary wave state contains
more, with larger amplitudes, a signature of the wavemaker importance in the
generation by sinusoidal forcing of integrable turbulence. Indeed, would the
wavemaker make no difference, integrability would impose the same number and
amplitudes of solitons at the beginning and in the stationary wave state. On
the other hand, they emphasize the possible role of the first soliton train.
Indeed, the soliton amplitudes of this first train are smaller than the
subsequent ones, but they travel on the rest level (same as the mean level $h$
in our experiments). As it appears on Fig. 5 the other soliton trains
propagate on a lower reference water level and with adverse velocity [53, 29].
The difference $\Delta h$ between mean water level and the reference water
level roughly corresponds to the mass of the solitons contained in one forcing
wave wave length divided by the wave length $l$. As [53] showed the velocity
$c$ of a soliton propagating on a reference level different from the rest
level is,

$c=\sqrt{g\,h}\,\left(1+\frac{1}{2}\frac{a}{h}-\frac{3}{2}\frac{\Delta
h}{h}\right).$ (20)

Consequently the very first train travels faster, it is not synchronized with
the others and can be amplified by the wavemaker. In this case when large
enough, the first soliton induces large phase shifts that in turn can
disynchronize other solitons that can be either amplified or attenuated by the
wavemaker. This chain of interactions triggers the randomization of the wave
motion by shuffling solitons that initially were ordered by decreasing
amplitudes. Since phase shifts between soliton increase with amplitude and
thus with the nonlinear $a/h$ parameter it conceivably explains why case (c)
disorganizes more rapidly than (b). The first soliton train acts as a catalyst
for the integrable turbulence.

##  VI Conclusion

Our experimental set-up differs from the integrable framework of Zabusky &
Kruskal [31] on various points. The most obvious is dissipation that imposes
some continuous energy flux input for the wave motion to possibly reach a
statistically stationary wave regime. However, the time scales of dissipation,
well represented by the Keulegan law, are much larger than those involving
soliton interactions which suggest that integrability still holds locally.
Indeed, we show that once a soliton gas is formed dissipation slowly alters
the amplitude distribution but not to the point where it would be obliterated
in a flume length propagation time.

The second difference lies of course in the finite flume length and the
reflecting behavior of both ends of the flume. We show that it does not only
allow for bi-directionnal integrable turbulence as described by the Kaup-
Boussinesq equations [54, 55] but also produces desynchronization of the waves
with the wavemaker motion.

Finally the third difference and probably the most important for the
generation of integrable turbulence is the wavemaker feedback on wave
amplification and attenuation. We emphasize that the route to a random
integrable turbulence wave motion strongly depends on the degree of non-
linearity $a/h$ of the wavemaker motion since the energy flux input by the
wavemaker is proportional to $(a/h)^{2}$.

To characterize if a given wave motion state is close to integrable turbulence
or not, we suggest different metrics. The nondimensionnal wavemaker Ursell
number and nonlinear parameter $a/h$ are relevant numbers to delineate
integrable turbulence from other regimes. An experimental coverage of the
($U$, $a/h$) plane indicates that integrable turbulence in our settings will
be sustained if $a/h>0.07$ and $U>0.2$. These bounds are approximate and
should be confirmed with more experiments. The mixing index $I_{m}$ that
characterizes the shape of the power density spectrum is used to assess if
wave periodicity has disappeared in the final wave motion state, one of the
signatures of integrable turbulence. We sustain that, alike numerical
simulation based on the KdV equation, kurtosis increases quasi linearly with
skewness. This needs to be further investigated in other conditions, but such
property may be useful in the analysis of oceanic wave data. Finally the
periodic scattering transform (PST) is used to precisely characterize the
spectral content of the empirical integrable turbulence. This transform is
unavoidable to determine the solitonic and radiative content of a given regime
and thus is the only tool to supply soliton amplitude distributions and fully
characterize wave motion random states.

The difference between unidirectional and bi-directionnal integrable
turbulence has not been addressed in the present study. The Kaup-Boussinesq
equations [54] offer an integrable framework to describe bi-directionnal
integrable turbulence in which the interactions between counter-propagating
solitons generate pulses than the sum of the amplitude of the two solitons
before interaction [56, 57] and also phase shifts. Nonetheless the approach
used in the present study takes advantage in the fact that, for weakly non-
linear solitons in an integrable turbulence, the inverse scattering problem
for the Kaup-Boussinesq equations can asymptotically be decomposed in left-
going and right-going KdV inverse scattering problems [54]. There are some
indications in the literature of IST techniques for Kaup-Boussinesq [58].
However, a practical periodic IST is yet to be developed and would require
extensive numerical developments. In future work different experimental
boundary conditions could be considered such as absorbing conditions at the
end wall with re-injection of the outgoing waves at the wavemaker
simultaneously with the generation of new wave trains conducive to a one way
integrable turbulence. While easy to implement numerically we foresee some
technical difficulties to do it on a real-time basis.

###### Acknowledgements.

We thank S. Randoux and P. Suret (PhLAM, Université de Lille) for the fruitful
discussions during I. Redor’s PhD. This project has received funding from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation program (Grant No. 647018-WATU). It was also partially
supported by the Simons Foundation through the Simons collaboration on wave
turbulence.

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