Emergence of Peregrine solitons in integrable turbulence of deep water gravity waves

We study experimentally the early stages of integrable turbulence of unidirectional deep water gravity waves. By generating partially coherent waves in a $148\text{−}\mathrm{m}$-long wave flume, we observe the emergence of high-amplitude structures formed by nonlinear focusing, commonly referred to as rogue waves. This work confronts the experiment with two recent results obtained in the framework of the nonlinear Schrödinger equation (NLSE), namely that (i) these structures can be locally fitted by a Peregrine soliton and (ii) their emergence leaves a visible trace on the evolution of statistical parameters such as kurtosis. We show that (i) yields accurate results as long as the wave steepness remains moderate, whereas (ii) is very robust and remains valid beyond the assumption of integrability. Numerical simulations of the NLSE and of the fully nonlinear dynamical equations are also performed to support these results.

Figure 1

Spatiotemporal diagram of the wave envelope amplitude $\left|A\right(z,\tau \left)\right|$ ($\epsilon =0.057$) inferred from the 20 probes, measurements then being stacked vertically. Local maxima are circled with a dashed line. The time series of the surface elevation $\eta (z,\tau )$ and of the extracted envelope $\left|A\right|(z,\tau )$ are also reported for the first and tenth probes ($z=6\mathrm{m}$ and $z=60\mathrm{m}$).

Figure 2

Statistical properties of the surface elevation along the flume for $\epsilon =0.057$ (left), $\epsilon =0.076$ (center), and $\epsilon =0.094$ (right). The kurtosis measured experimentally is compared to two numerical simulations (NLS and HOS) introduced in Sec. 3, and to the experimental histograms of local maxima (all of them in orange bins, only the ones successfully fitted to a Peregrine soliton in black bins).

Figure 3

Power spectral density (PSD) of the surface elevation $\eta (z,t)$ for the smallest steepness $\epsilon =0.057$ at various locations $z$ along the flume. Note the spectral broadening at $z=84\mathrm{m}$. Similar overshoots of the spectral bandwidth are observed for the two larger steepnesses.

Figure 4

Fit of the top left circled local maxima of Fig. 1 to a Peregrine soliton [Eq. (8)]. The distance to the wave maker is $z=78\mathrm{m}$ and the initial steepness is $\epsilon =0.057$. Both the direct signal $\eta (z_{\max },t)$ (left) and the phase and amplitude of the complex envelope $A(z_{\max },t)$ (right) are reported, the dashed lines corresponding to the best fit.

Figure 5

Every local maxima of the wave envelope is fitted to a Peregrine soliton (see, e.g., four of them identified in Fig. 1 and a fit of one of them is shown in Fig. 9). The fraction of successful fits is reported as a function of the initial steepness, evidencing the decrease with $\epsilon$ of the PS as an accurate local model of gradient catastrophe regularization. Negative and positive error bars correspond to the same measure performed with the right-hand side of Eq. (9) being respectively 0.2 and 0.3 instead of $1/4$.