Experimental Observation and Theoretical Description of Multisoliton Fission in Shallow Water

We observe the dispersive breaking of cosine-type long waves [Phys. Rev. Lett. 15, 240 (1965)] in shallow water, characterizing the highly nonlinear “multisoliton” fission over variable conditions. We provide new insight into the interpretation of the results by analyzing the data in terms of the periodic inverse scattering transform for the Korteweg–de Vries equation. In a wide range of dispersion and nonlinearity, the data compare favorably with our analytical estimate, based on a rigorous WKB approach, of the number of emerging solitons. We are also able to observe experimentally the universal Fermi-Pasta-Ulam recurrence in the regime of moderately weak dispersion.

Figure 1

Fission from three periods of a cosine wave with ${\eta }_0=1.84\mathrm{cm}$ in $h=20\mathrm{cm}$ water (run $B$ in Table 1). In order to follow the evolution of the same initial period, different time intervals are displayed. Open dots in (f), (g) stand for soliton amplitudes computed from IST analysis of data sets [see text and Fig. 3], recorded at 55 and 65 m.

Figure 2

Wave profiles: (a)–(c) run $C$, $h=15\mathrm{cm}$, ${ϵ}^2=0.0181$; (d)–(f) run $D$, $h=10\mathrm{cm}$, ${ϵ}^2=0.0107$.

Figure 3

(a) Half-trace of monodromy matrix $T(\lambda )$ vs spectral parameter $\lambda$ at ${ϵ}^2=0.029$ (ZK regime), comparing the ideal monochromatic case $u_0=\cos (\tau )$, with normalized experimental data, run $B$, $z_6=55\mathrm{m}$ [Fig. 1]. ${\lambda }_{1,\dots ,21}$ are the band edges; $N_s=8$ solitons correspond to the first eight bands on the left of ${\lambda }_S$ (see text for its definition). This sets ${\lambda }_{\mathrm{ref}}$ to ${\lambda }_{17}$ ($2N_s+1=17$). (b) Real-world soliton amplitudes $2{\eta }_0[{\lambda }_{17}(z_j)-{\lambda }_{2n}(z_j)]$, $n=1,\dots ,8$, obtained from the $p$-IST analysis, as reported in (a), of recorded elevation $\eta (t,z_j)$ at gauge distances $z_j$ [data at $z=0$, obtained from $u_0=\cos (\tau )$, are reported for reference].

Figure 4

Number of solitons $N_s$ vs dispersion smallness ${ϵ}^2=(2{\omega }^2{h}^2/3g{\eta }_0)$, as extracted from $p$-IST of experimental data at $z_6=55\mathrm{m}$. Equal symbols and colors indicate data sets recorded at same depth $h$. The solid line stands for the WKB result in Eq. (4), $\kappa =0.01$, with its envelope (dashed line) emphasizing leading order dependence $N_s\sim 1/ϵ$.

Figure 5

Recurrence observed for run $E$, $h=40\mathrm{cm}$: (a) Fourier mode evolution from data measured at the eight gauge locations $z_j$; (b)–(d) Wave profiles at $z_2=15$, $z_7=65$ (recurrence), $z_8=75\mathrm{m}$ (new cycle of fission).