Group-wave resonances in nonlinear dispersive media: The case of gravity water waves

The dynamics of coherent nonlinear wave groups is shown to be drastically different from the classical scenario of weakly nonlinear wave interactions. The coherent groups generate nonresonant (bound) waves which can be synchronized with other linear waves. By virtue of the revealed mechanism, the groups may emit waves with similar or different lengths, which propagate in the same or opposite direction.

Figure 1

The maximum displacement $\eta$ as the function of time (black line) and three selections by the windows $M$ (the red areas), for the case $\epsilon =0.07, L/\lambda =6$.

Figure 2

The space-time Fourier transform amplitudes ${log}_{10}S(k,\omega )$ are shown by color (a) well before the first focusing ($t\approx 3100T_0$), (b) at the moment of focusing ($t\approx 3600T_0$), and (c) when demodulated ($t\approx 4700T_0$), for the case $\epsilon =0.07, L/\lambda =6$. The white curves show the Fourier transform amplitudes ${log}_{10}{S}_{\omega +}$ (right), ${log}_{10}{S}_{\omega -}$ (left), and ${log}_{10}{S}_k$ (upper). The cyan dotted lines plot the dispersion curves. See Supplemental Material [11].

Figure 3

The Fourier transform amplitudes (a) at the moment of focusing ($t\approx 1600T_0$) and (b) well after ($t\approx 2000T_0$), for the case $\epsilon =0.11, L/\lambda =4$. The notations are similar to Fig. 2. See Supplemental Material [11].

Figure 4

The Fourier transform amplitudes (a) at the moment of focusing ($t\approx 1500T_0$) and (b) well after ($t\approx 1900T_0$), for the case $\epsilon =0.146, L/\lambda =3$. The cyan dashed lines in (b) plot the modified dispersion curve. The other notations are similar to Figs. 2 and 3. See Supplemental Material [11].