Diffraction of deep-water solitons

Abstract

Solitons are localized nonlinear wave packets that propagate without spreading because nonlinearity balances dispersion. Their robustness is well understood in effectively one-dimensional systems, but introducing additional spatial dimensions is generally expected to destabilize them or destroy their coherent character. Here we experimentally investigate how deep-water gravity-wave solitons behave when a controlled transverse degree of freedom is introduced through diffraction. Using a large-scale water-wave facility, we generate solitonic wave packets whose transverse structure is imposed across a segmented wavemaker through either a sharp slit or a smooth Gaussian apodization. The resulting two-dimensional wave fields are measured with high spatial resolution. Diffraction reshapes the transverse profile of the wave packet while its longitudinal dynamics retain the characteristic features of a soliton. Nonlinear spectral analysis confirms that the solitonic content is preserved along the direction of propagation, whereas the transverse evolution follows the linear Fresnel laws of diffraction. These observations reveal an unexpected coexistence of nonlinear soliton dynamics and classical wave diffraction.

Figures (5)

• Figure 1: Experimental set-up. Top: Schematic representation (not to scale) of the $3$D water tank (top view) used in the experiments. The transverse profile of the generated waves can be carefully shaped using 48 computer-controlled segmented wavemakers placed along the y axis, at $x=$ 0 m. Horizontal red bars: $45$ wave elevation probes are placed at discrete propagation distances $x = 1$–$25\,\mathrm{m}$ and transverse positions $y \in [0.1, 29.64]\,\mathrm{m}$ with a non-uniform spacing, as indicated by the probe array. The wavelength of the carrier wave is approximately $\lambda_c\simeq 1.3$ m. Bottom: Image of a diffracting wave with an opening of $D=1.2$ m (i.e. $2$ flaps out of $48$), see Movies S1 and S2.
• Figure 2: Measured surface elevation $\eta(y,t)$ of a soliton of steepness $\epsilon=k_0a=.097$ measured by the wave probes at $L=20$ m for three different slit openings (see white rectangles) $D=30$(a),15(b) and 10(c) m. For the largest opening, (a), we observe the classical profile of a 1D NLSE soliton [\ref{['eq:soliton']}]. Decreasing the aperture width, as seen in (b) and (c) reduces the transverse ($y$) size of the soliton, however we still observe a coherent structure in the basin center. Additionally, we observe the appearance of distinct minima and maxima in the transverse profile, consistent with classical wave diffraction.
• Figure 3: Transverse structure of diffracting deep-water solitons. Amplitude and IST profiles with comparison with HNLSE and Helmholtz diffraction. The columns correspond to different aperture widths $D=1,\,10,\,15,\,20,$ and $30~\mathrm{m}$ (from left to right), measured at a fixed propagation distance $L=20~\mathrm{m}$. (a1--a5) Transverse envelope amplitude $|A(y)|$. Black dots: experimental data ($\epsilon = 0.097)$; solid blue curves: numerical simulations of the HNLSE \ref{['eq:nls_2d']}; dashed orange curves \ref{['eq:fresnel']}: linear Helmholtz diffraction by a rectangular slit of width $D$ evaluated for the carrier wavenumber $k_0$. The Helmholtz predictions are rescaled by a single multiplicative factor to match the experimental peak amplitude in each panel. (b1--b5) IST spectra extracted from the longitudinal wave field and represented as function of the transverse coordinate $y$. Blue points correspond to imaginary part of the discrete eigenvalue, $\mathrm{Im}(\zeta)$, which characterizes the soliton amplitude. The presence of discrete eigenvalue across $y$ indicate the solitonic nature of the wave packet. (c1--c5) Steepness-normalized transverse amplitude $|A(y)|/\epsilon$ for different soliton amplitudes, $a\in[0.37,2.61]\, \mathrm{cm}$, generated with the same aperture $D$. For each run, the experimental curves (markers) and the associated HNLSE profiles (solid curves) are shown for several steepness values $\epsilon=k_0 a \in [0.02,0.13]$ (the color encodes the soliton steepness, as indicated by the colorbar on the right).
• Figure 4: (a) Measured surface elevation $\eta(y,t)$ of a solitonic Gaussian beam with $\epsilon \approx 0.072$ and waist $W_0\approx 6.43$ m. The beam is well localized in the transverse direction, and unlike the slit-diffracted soliton, we observe a much more regular wavefront due to the Gaussian apodization (white bar). (b) Transverse cut of a reference soliton at $x=20~\mathrm{m}$, showing the measured envelope amplitude $|A(y)|$ (blue, solid) together with a Gaussian fit (black, dashed), and the corresponding unwrapped transverse phase profile $\phi(y)$ (green, solid) with a parabolic fit (black, dotted), from which the wavefront radius of curvature is extracted. The arrow indicates the fitted transverse waist $W$ of the Gaussian envelope. (c) Measured waist $W^{\mathrm{exp}}$ as a function of the theoretical prediction $W^{\mathrm{th}}$ (\ref{['eq:gaussbeam']}) for different initial waists $W_0$. Inset : Measured radius of curvature $R^{\mathrm{exp}}$ versus the theoretical radius $R_{\mathrm{G}}$ (\ref{['eq:gaussbeam']}). In panel (c) and its inset, filled symbols correspond to experiments and open symbols to HNLSE simulations. Marker shapes encode different initial waists ($W_0\approx \diamond 3.87,\, \triangle 5.15,\, \square 6.43,$ and $\circ7.73~\mathrm{m}$), while the color scale on the right indicates the soliton steepness $\epsilon = k_0 a$.
• Figure 5: (a) Absolute value of the soliton wavefront radius of curvature $|R_S|$, extracted from the carrier-wave front, as a function of the rescaled variable $L + D^2/(\pi^2 \lambda_0 \sqrt{\epsilon})$, where $L$ is the propagation distance, $D$ the transverse aperture (slit) width, $\lambda_0$ the carrier wavelength, and $\epsilon = k_0 a$ the steepness. Full (open) symbols denote experiments (HNLSE simulations). Triangles correspond to $L=20\,\mathrm{m}$ and diamonds to $L=35\,\mathrm{m}$. The color encodes $\epsilon$. Error bars indicate the standard deviation over all fits of the individual carrier-wave oscillations. The black dashed line (slope $1$) highlights the empirical collapse of \ref{['eq:scaling']}. (b) (log--log axes): Curvature $|R|-L$ versus $D$. Circles ($L=20\,\mathrm{m}$) and squares ($L=35\,\mathrm{m}$) show Gaussian-apodized beams ($R_G$) extracted from quadratic phase fits (full: experiments; open: HNLSE simulations), with $D= 2W_0 \sqrt{\ln 2}$ for five different steepness values. Black dash-dotted line correspond to \ref{['eq:gaussbeam']} with $L=35$ m and the blue one to the $D^2$ scaling expected from \ref{['eq:scaling']} with $\epsilon=0.1$. Error bars for these points originate from the uncertainties of the quadratic phase fits, as estimated from the covariance matrix of the fit parameters.