Experimental Evidence of a Hydrodynamic Soliton Gas

We report on an experimental realization of a bidirectional soliton gas in a 34-m-long wave flume in a shallow water regime. We take advantage of the fission of a sinusoidal wave to continuously inject solitons that propagate along the tank, back and forth. Despite the unavoidable damping, solitons retain their profile adiabatically, while decaying. The outcome is the formation of a stationary state characterized by a dense soliton gas whose statistical properties are well described by a pure integrable dynamics. The basic ingredient in the gas, i.e., the two-soliton interaction, is studied in detail and compared favorably with the analytical solutions of the Kaup-Boussinesq integrable equation. High resolution space-time measurements of the surface elevation in the wave flume provide a unique tool for studying experimentally the whole spectrum of excitations.

Figure 1

Two-soliton interaction from experiment. (a) Head-on interaction of large amplitude solitons in a space-time representation. The wave maker is located at $x=0\mathrm{m}$. Small dispersive waves radiated from the collision can be seen in the background, a sign of small departure from integrability. (b) Snapshots of head-on interaction of small amplitude solitons ($a/h=0.1$) traveling in opposite directions. Red, 2 s before the collision; blue, at the collision; green, 5 s after the collision. Black dashed lines show the solution of the KB equations (see text for details). (c) Collision of solitons traveling in the same direction (overtaking interaction), same color code as in (b).

Figure 2

(a) Space-time representation of the soliton gas. The horizontal scale is the same for all panels. Solitons traveling in both directions are visible. Many head-on collisions can be seen. The dashed line is the long wave phase velocity $c_0=\sqrt{g{h}_R}$, with $h_R$ the reference level of soliton propagation. (b) Part of the signal propagating away from the wave maker (to the right) obtained by the Hilbert transform. (c) Extraction of the surface elevation profile corresponding to the dashed line in (b) at $t=11.5\mathrm{s}$. Magenta, fit of a single soliton solution to an isolated pulse. The horizontal dotted line is the corresponding reference level $h_R$. Red, the surface elevation at $t=10.3\mathrm{s}$ corresponding to dashed line in (b). Green, surface elevation at $t=13.5\mathrm{s}$ corresponding to the short dashed line in (b). These two curves correspond to pre- and postcollision states of a collision that occurs at $t=11.5\mathrm{s}$ and $x=19\mathrm{m}$. They illustrate an example of overtaking soliton interaction in the soliton gas.

Figure 3

(a) $E(k,\omega )$ spectrum of the soliton gas. Dashed curves, $aa$ is the dispersion relation of linear waves $\omega =\sqrt{gk\tanh (kh)}$. $bb$ is the relation $\omega =c_{0}k$. $cc$ shows the part of the spectrum related to transverse waves. (b) Frequency spectrum of the wave elevation average spatially over the locations accessible to the cameras. Black line, full signal. Red and blue lines, signal propagating to the right and to the left, respectively. The vertical dashed line corresponds to the forcing frequency at 0.6 Hz.