Gray solitons on the surface of water

The dynamics of surface gravity water waves can be described by the self-defocusing nonlinear Schrödinger equation. Recent observations of black solitons on the surface of water confirmed its validity for finite, below critical depth. The black soliton is a limiting case of a family of gray soliton solutions with finite amplitude depressions. Here, we report observations of gray solitons in water waves, thus, complementing our previous observations of black solitons.

Figure 1

(a) An example of a gray soliton for $a=1$, $\chi =0$, and $\vartheta =\frac{\pi }{6}$. (b) Gray soliton amplitude depressions at $T=0$ for $a=1$, $\chi =0$, and $\vartheta \in \{\frac{\pi }{3},\frac{\pi }{6},0\}$.

Figure 2

Simple geometric representation of the gray soliton on the complex plane of $\psi$. At any fixed $T$, the solution is represented by the points on the line $AB$. The points $A$ and $B$ represent the plane waves at the left- and at the right-hand sides of the soliton. The point $C$ corresponds to the point of minimal soliton amplitude. Parameter of the family $\vartheta$ is the angle $A$ (or $B$). The phase shift between the left- and right-hand sides of the soliton is given by the central angle ($\pi -2\vartheta$).

Figure 3

Evolution of a gray soliton along the wave flume with $\vartheta =\frac{\pi }{6}$, for carrier amplitude of $0.02$ m, a water depth of $h=0.25$ m and $kh=0.8$. The measurements are shifted in time in order to cancel the combined soliton and the wave group velocity, which are of $1.22$ ms${}^{-1}$, and keep the depression of the gray soliton at the central position.

Figure 4

Evolution of a gray soliton with $\vartheta =\frac{\pi }{14}$ and carrier amplitude of $0.02$ m, a water depth of $h=0.25$ m and $kh=0.9$. The measurements are shifted in time, as in Fig. 3, with respect to the velocity value of $1.13$ ms${}^{-1}$.

Figure 5

(a) Superposition of the envelopes of the observed gray soliton, shown in Fig. 3, measured by the first wave gauge (solid blue line), and the last wave gauge (black dash-dotted line) with the theoretical envelope (red dashed line). (b) Same as in (a), but for the gray soliton shown in Fig. 4.

Figure 6

Evolution of a black soliton ($\vartheta =0$) along the wave flume for carrier amplitude $0.075$ m. The water depth is $h=0.2$ m and $kh=0.6$. Here, the group velocity of the background waves, which correspond to $1.18$ ms${}^{-1}$, has been canceled to keep the black soliton in the center.