\section*{Methods}



\subsection*{One-dimensional Nonlinear Schr\"odinger equation, variables and parameters}
In the context of uni-directionnal deep water waves, 1DNLSE reads:
 \begin{equation}
i\frac{\partial \psi}{\partial z}= \frac{1}{g}\frac{\partial^2 \psi}{\partial t^2}+\gamma |\psi|^2\psi, \label{eq:NLSE}
\end{equation}
where $z$ is the propagation distance, $t$ is the time measured in the frame co-propagating a the group velocity $c_g=g/(2 \omega_0)$  evaluated at the carrier frequency $f_0=\omega_0/(2\pi)$. In the limit of infinite depth, the third order nonlinear coefficient is:
\begin{equation}
\gamma=k_0^3
\end{equation}
where $k_0=\omega_0^2/g$ is the modulus of the wavevector of carrier wave. The surface elevation $\eta(t,z)$ is related at the leading order to the  slowly varying complex envelope $\psi$ as:
\begin{equation}
\eta(t,z)=\frac{1}{2}\left( \psi(t,z)e^{i (k_0z -\omega_0 t)}+c.c.\right); \label{etaA}
\end{equation}

From (\ref{etaA}) the following relation can be derived :
\begin{equation}
\langle|\psi|^2\rangle=2\langle\eta^2\rangle=2 \sigma^2, \label{amplitudesurface}
\end{equation}
where $\langle...\rangle$ implies averages over time and $\sigma^2$ is the variance of the rapidly oscillating wave field.

In order to design the experiment, it is useful to introduce  a linear and a nonlinear propagation length  as follows:
\begin{equation}
z_{lin}=\frac{g}{\Delta \omega^2}\;\;\ {\rm and}\;\;\;\;
z_{nlin}=\frac{1}{\gamma P_0},
\label{eq:zlznl}
\end{equation}
where $\Delta \omega=2\pi \Delta f$ is a typical spectral bandwidth  and $P_0=\langle|\psi_0|^2\rangle$ is the average value of the envelope square both calculated at $z=0$.


The degree of nonlinearity of the wave propagation is given by the parameter:
\begin{equation} 
\Gamma=\frac{z_{lin}}{z_{nlin}}=\frac{\gamma g P_0}{\Delta \omega^2}.
\end{equation}

In the context of ocean waves, $\Gamma=BFI^2$ where BFI is the Benjamin-Feir Index \cite{Janssen:03,Onorato:01}. It has been shown that initial conditions with a large value of $\Gamma$ will eventually lead to the formation of rogue waves characterized by heavy-tailed probability density function of the surface elevation~\cite{Walczak:15,Koussaifi:18,Suret:16,Tikan:18}.

BFI index can be expressed as follows:
\begin{equation}
BFI = \frac{\epsilon}{(\Delta f /f_0)},    
\end{equation}
where $\epsilon= k_0 \sqrt{2} \sigma$ is the wave steepness, and $\Delta f$ and $f_0$ are average spectral width and central frequency of the initial wave packets.  
\\

\subsection*{Resonances in the 1DNLSE}

The non trivial resonances are forbidden in integrable turbulence~\cite{Suret:11}. We consider the third order nonlinear interaction of four monochromatic waves $i=1,2,3,4$ of pulsation $\omega_i$ and wavenumber $k_i(\omega_i)$ in a unidirectional dispersive media. In this context, the resonances conditions of four wave mixing in the standard wave turbulence read $\omega_1+\omega_2=\omega_3+\omega_4$ and $k_1+k_2=k_3+k_4$ where $k_i=k(\omega_i)$. The linear dispersion relation of deep water waves and of the Eq.~({\ref{eq:NLSE}}) is: 

\begin{equation}
    k(\omega)=\frac{\omega^2}{g}
    \label{eq:kL}
\end{equation}

Exact resonances thus lead to:

\begin{equation}
     \omega_1(\omega_3-\omega_2)=\omega_3(\omega_3-\omega_2)
\end{equation}

{\it i.e.} $\omega_1=\omega_3$ and $\omega_2=\omega_4$ or $\omega_2=\omega_3$ and $\omega_1=\omega_4$.  

Exact resonances of non trivial interactions ($\omega_1\ne\omega_3$ and $\omega_1\ne\omega_4$) are thus forbidden.

\subsection*{Initial conditions in simulations and experiments: Partially coherent waves}

In the numerical simulations and in the experiments, the initial conditions are partially coherent waves~\cite{Randoux:16}. The slow varying amplitude of the initial condition at $z=0$ reads:

\begin{equation}
    \psi(t,z=0)=A_0 \sum_{l=-N/2}^{+N/2} e^{-\frac{1}{2}(f_l/\Delta f)^2} e^{i \,2 \pi\,f_l\,t} e^{i\phi_l}
\end{equation}

where $f_l=l/T_{max}$, $T_{max}$ is the temporal duration of the experiments and $\phi_l$ are independently and randomly distributed over $[0,2\pi]$.


 Note that the statistical characteristics of partially coherent waves are very different than plane waves initially perturbed by noise which is also investigated in \cite{Leisman:EDR:19}. For a comparison between the two cases, refer {\it e.g.} to \cite{Copie2020}.



\subsection*{Numerical Simulations}
Numerical simulations of Eq.\ref{eq:NLSE} are realized using step-adaptive high order Runge-Kutta method. We construct different initial conditions using the random phase approach where a uniformly distributed phase is added to every Fourier component of a Gaussian spectrum with $\Delta f=$ 0.2 Hz. Typical temporal windows used in the numerical simulations correspond to 100 seconds.  Three parameters of simulations depend on the value of the steepness : the number of points $N$, the length of propagation $L_{max}$ and the number of realizations $N_{sample}$. We separate the numerical studies into three ranges of steepness $\epsilon$:



\begin{tabular}{llll}
    & $\;\;\;N\;\;\;$ & $L_{max}(m)$ & $N_{sample}$ \\
  $\;\;\;\;\;\;\;\;\;\;\;\epsilon\le 0.05$ & $\;\;2048\;\;$ & $\;\;\;500$ & $\;10000$ \\
  $0.05<\epsilon\le 0.09$ & $\;\;2048\;\;$ & $\;\;\;500$ & $\;\;\;\;500$ \\
  $0.09<\epsilon\le 0.19$ & $\;\;1024\;\;$ & $\;\;2000$ & $\;\;\;\;\;100$ \\
\end{tabular}\\

In order to reconstruct numerically NDR, we multiply the spatiotemporal diagram by a Super-Gaussian window with power 15 along z direction avoiding thereby undesirable effects related to the Fourier analysis of non-periodic signals.

\subsection*{Resolution of the measurement of $k$}

 The key point in our approach is to remove the carrier wave before computing the NDR. This allows us to reveal the details of the NDR of the slowly varying envelop. Note that we only use 20 gauges in the water tank. They are separated by $6$~m and the maximum measurable wavevector is around $1.05$~m$^{-1}$. As a consequence, contrary to Taklo {\it et al.} who use 384 probes, we do not resolve the wavevectors of the carrier wave $2\pi/\lambda_0\simeq5.3$~m$^{-1}$ and of the harmonics. Our strategy enables the accurate  measurement of the NLDR of the slow varying envelop of the wave by using only 20 probes. This provides an effective measuring range of $120$~m with a resolution of the measurement of the $k-$spectrum  of $\Delta k_{min}=2\pi/120$~rad~m$^{-1}$. Note finally that, in~\cite{Taklo2015}, the accuracy of the  measurement of $k$ given by the length of the water tank is $\Delta k_{min}/k_0=0.014$ while our setup enables an accuracy of $\Delta k_{min}/k_0 < 0.01$. 

In order to measure the averaged spectra and NDR, we use 3,6 and, 3  experimental runs with a duration of 512~s for $\Gamma=0.12$, $0.33$, $0.65$, respectively. One run of 128~s have been used for $\Gamma=6.18$.

Note that in NLS and HOS simulations, the chosen lengths of propagation depend on the parameters and vary typically from 300 to 500~m. The uncertainty of measurement of $k_M$ and $\Delta k$ is therefore significantly lower in simulations than in experiments.

\subsection*{Evaluation of the full width at half maximum $\Delta k$ of the position of the maximum $k_M$}

In Fig. 4, we report the evaluation of the full width at half maximum $\Delta k$ and of the position of the maximum $k_M$ of the function $f(k)=|\widetilde{\psi}(k,\omega=0)|^2$ in 1DNLSE, HOS simulations and in experiments. The accuracy of the measurement of $\Delta k$ and $k_M$ is limited both by  the discretization of $k$ (see above the uncertainty $\Delta k_{min}$) and by the random fluctuations of $f(k)$. In order to overcome these difficulties, when it is appropriate, we evaluate $\Delta k$ and $k_M$ by using best fitting procedure with Lorentzian function.



\newpage
\section*{Supplementary Material}


The purpose of this Supplementary Information is to provide some mathematical and numerical details on the High-Order Spectral (HOS) method that is utilized as a reference in the manuscript. Simulation results (spatio-temporal diagrams and NDR) obtained with the model are also presented.

\subsection*{High-Order Spectral method}

One-dimensional Nonlinear Schr\"odinger equation (Eq. (3) of manuscript) is established under the assumptions of weak nonlinearity of the wave field as well as the narrowbandedness of its energy content. Different degrees of nonlinearity ($\Gamma$ parameter) have been tested during the wave tank experiments reported in the manuscript. Then, it appears of great interest to have a high-fidelity numerical model able to replicate the experiments.

This is achieved thanks to the direct numerical simulation of the Euler's equation solved using the High-Order Spectral (HOS) method \cite{dommermuth1987high,west1987new}. The present numerical results have been obtained with the open-source solver HOS-NWT \cite{HOS-NWT}. This model solves the spatio-temporal evolution of the free surface elevation $\eta(z,t)$ in a so-called numerical wave tank. It reproduces all the physical features of the experimental (physical) facility including i) the generation of waves thanks to a wave maker and ii) the absorption of those waves when they reach the opposite wall. More details on the HOS-NWT numerical model as well as different validations performed against wave tank experiments can be found in \cite{bonnefoy2010time,ducrozet2012modified}.

This digital twin is utilized as a complement to the experiments. In the present study, the main advantage of the numerical solution is that it allows to overcome the limitation associated with the finite extent of the physical wave tank (and the associated measuring range $L_m=120~m$). As long as the spatial discretization is kept identical, the numerical wave tank can be of arbitrary size. Increasing the size results in a more accurate resolution $\Delta k_{min}$ for the estimation of wave number $k$, essential to the evaluation of the NDR and its properties (Fig.  4 of the manuscript). \\

\begin{figure*}[!t]
  \includegraphics[width=0.73\linewidth]{File_C12_C10_series_04_run_24_HOS_eps_0p04.png}
  \caption{\textbf{Numerical simulation using high-order spectral methods $\Gamma$ = 0.05,
 $\epsilon$ = 0.04.} \textbf{(top)}  Spatiotemporal diagram of the wave envelope amplitude $|\Psi(t,z)|$ evolution over the 531 m of the numerical water tank having 177 probes. \textbf{(bottom)} Nonlinear dispersion relation $|\widetilde{\psi}(\omega,k)|^2$ reconstructed from the evolution of the complex wave envelope.}
  \label{fig1si}
\end{figure*}

\subsection*{Numerical Simulations}




In a wave tank environment, numerical and physical experiments start from temporal initial conditions $\eta(z,t=0)$ at rest. Waves are generated thanks to a wave maker that imposes the spatial initial condition $\eta(z=0,t)$. The HOS digital twin of the \'Ecole Centrale de Nantes water tank facility (Fig. 2.a of original manuscript) uses the exact same wave maker motions than the experimental ones. This allows the direct deterministic comparison between the experimental and numerical wave probes measurements for validation purposes.



Numerical parameters are chosen after a careful convergence study that ensures the accuracy of the numerical solution. The time integration, achieved thanks to a step-adaptive Runge-Kutta method, is controlled by a tolerance parameter chosen as $10^{-8}$ for the present long-time integration. An HOS order of nonlinearity set to $5$ ensures an accurate numerical solution for all configurations tested. Regarding the spatial discretization, the total length of the numerical domain is set to $L_x=560~m$, discretized with $N_x=12288$ points/modes free of aliasing errors.\\


The wave conditions simulated cover a wide range of nonlinearity with $\Gamma=\left[ 0.05 ; 0.12 ; 0.33 ; 0.48 ; 0.65\right]$. The spatiotemporal diagram of the wave enveloped amplitude $|\Psi(t,z)|$ as well as the corresponding NDR $|\widetilde{\psi}(k,\omega)|^2$ are presented in Figures \ref{fig1si} to \ref{fig5si} with an increasing level of nonlinearity. 

Taking into account the evolution of the nonlinear scale $z_{nlin}$ with $\Gamma$ (Eqs. (7) and (8) of manuscript), the analysis is performed on a length of $531~m$ for $\Gamma=0.05$ and $\Gamma=0.12$ in Figs. \ref{fig1si} and \ref{fig2si} respectively. With an increased nonlinearity $\Gamma=\left[ 0.33 ; 0.48 ; 0.65\right]$, the analysis is conducted on a shorter length of $378~m$. Similarly to the experimental facility, the numerical results use wave gauges that are equally spaced in the computational domain every $6~m$.\\


Overall, the HOS numerical simulations confirm observations in the physical wave tank. As expected, comparing left column of Fig. 1 in the manuscript and Fig. SI \ref{fig2si}, for small values of $\Gamma$ 1DNLSE and HOS results are very similar with a small departure from linear dispersion relation. Then, an increased nonlinearity is associated with coherent structures that are more visible in the $(t,z)$ diagram, see Fig. SI \ref{fig5si} for instance. However, compared to the 1DNLSE results (Fig.1 right column of manuscript), the corresponding straight lines are less frequent in the Euler's simulations. In addition, the evolution of those structures is clearly subjected to a negative speed when taking into account high-order nonlinear and dispersion effects. The corresponding frequency down-shift is also clearly visible in the NDR.

\begin{figure*}[!h]
  \includegraphics[width=0.73\linewidth]{File_C12_C10_series_04_run_10_HOS_eps_0p06.png}
  \caption{{Numerical simulation using high-order spectral methods $\Gamma$ = 0.12,
 $\epsilon$ = 0.06.} \textbf{(top)}  Spatiotemporal diagram of the wave envelope amplitude $|\Psi(t,z)|$ evolution over the 531 m of the numerical water tank having 177 probes. \textbf{(bottom)} Nonlinear dispersion relation $|\widetilde{\psi}(\omega,k)|^2$ reconstructed from the evolution of the complex wave envelope.}
  \label{fig2si}
\end{figure*}


\begin{figure*}[!h]
  \includegraphics[width=0.73\linewidth]{File_C12_C10_series_04_run_09_HOS_eps_0p1.png}
  \caption{{Numerical simulation using high-order spectral methods $\Gamma$ = 0.33,
 $\epsilon$ = 0.1.} \textbf{(top)}  Spatiotemporal diagram of the wave envelope amplitude $|\Psi(t,z)|$ evolution over the 378 m of the numerical water tank having 126 probes. \textbf{(bottom)} Nonlinear dispersion relation $|\widetilde{\psi}(\omega,k)|^2$ reconstructed from the evolution of the complex wave envelope.}
  \label{fig3si}
\end{figure*}

\begin{figure*}[!h]
  \includegraphics[width=0.73\linewidth]{File_C12_C10_series_04_run_22_HOS_eps_0p12.png}
  \caption{{Numerical simulation using high-order spectral methods $\Gamma$ = 0.48,
 $\epsilon$ = 0.12.} \textbf{(top)}  Spatiotemporal diagram of the wave envelope amplitude $|\Psi(t,z)|$ evolution over the 378 m of the numerical water tank having 126 probes. \textbf{(bottom)} Nonlinear dispersion relation $|\widetilde{\psi}(\omega,k)|^2$ reconstructed from the evolution of the complex wave envelope.}
  \label{fig4si}
\end{figure*}


\begin{figure*}[!h]
  \includegraphics[width=0.73\linewidth]{File_C12_C10_series_04_run_20_HOS_eps_0p14.png}
  \caption{{Numerical simulation using high-order spectral methods $\Gamma$ = 0.65,
 $\epsilon$ = 0.14.} \textbf{(top)}  Spatiotemporal diagram of the wave envelope amplitude $|\Psi(t,z)|$ evolution over the 378 m of the numerical water tank having 126 probes. \textbf{(bottom)} Nonlinear dispersion relation $|\widetilde{\psi}(\omega,k)|^2$ reconstructed from the evolution of the complex wave envelope.}
  \label{fig5si}
\end{figure*}


 \clearpage 

\begin{acknowledgments}
This work has been partially supported  by the Agence Nationale de la
Recherche  through the LABEX CEMPI project (ANR-11-LABX-0007) and the ANR DYSTURB Project (ANR-17-CE30-0004),   the
Ministry of Higher Education and Research, Hauts de France council and
European Regional Development  Fund (ERDF) through the the Nord-Pas de
Calais Regional Research Council and the European Regional Development
Fund (ERDF) through the Contrat de Projets Etat-R\'egion (CPER
Photonics for Society P4S).  E.F. is thankful for partial support from the Simons Foundation/MPS No. 651463
\end{acknowledgments}

