Nonlinear shallow ocean-wave soliton interactions on flat beaches

Ocean waves are complex and often turbulent. While most ocean-wave interactions are essentially linear, sometimes two or more waves interact in a nonlinear way. For example, two or more waves can interact and yield waves that are much taller than the sum of the original wave heights. Most of these shallow-water nonlinear interactions look like an X or a Y or two connected Ys; at other times, several lines appear on each side of the interaction region. It was thought that such nonlinear interactions are rare events: they are not. Here we report that such nonlinear interactions occur every day, close to low tide, on two flat beaches that are about 2000 km apart. These interactions are closely related to the analytic, soliton solutions of a widely studied multidimensional nonlinear wave equation. On a much larger scale, tsunami waves can merge in similar ways.

Figure 1

A plot and a photograph of an X-type interaction. (a) A plot of an analytical line-soliton interaction solution of the KP equation at $t=0$ using (3) and (4). In this and the following plots, we picked the $k_i$ and $P_j$ to be qualitatively similar to the photograph in part (b). Here, $k_1=k_2=1/2$, $P_1=-P_2=2/3$ so $e^{A_{12}}\approx 2.3$. (b) Taken in Mexico on 31 December 2011; notice the large amplitude of the short stem. (c) A 3d plot of the solution in (a), which qualitatively agrees with (b); we only include one 3d plot because the density plots show the interaction behavior clearly.

Figure 2

A plot and photographs of a Y-type interaction. (a) $k_1=1/2$, $k_2=1$, $P_1=3/4$, $P_2=1/4$ so $e^{A_{12}}=0$. (b) Taken in Mexico on 6 January 2010. (c) Taken in California on 3 May 2012.

Figure 3

A plot and photographs of an X-type interaction with a longer stem. (a) $k_1=k_2=1/2$, $P_1=-1/4-{10}^{-2}$, $P_2=3/4$ so $e^{A_{12}}\approx 51$. (b) Taken in California on 2 May 2012 in shallower water than Fig. 1b. (c) Taken in California on 4 May 2012.

Figure 4

A plot and photographs of an X-type interaction with a very long stem. (a) $k_1=k_2=1/2$, $P_2=-P_1+{10}^{-10}=1/2$ so $e^{A_{12}}\approx 5\times {10}^9$. (b) Taken in Mexico on 28 December 2011 in shallower water than Fig. 3b. (c) Taken in California on 3 May 2012.

Figure 5

A plot and photographs of an X-type interaction, where the stem has a lower rather than a higher amplitude. (a) $k_1=1$, $k_2=1/2$, $P_1=1/2-{10}^{-7}$, $P_2=0$ so $e^{A_{12}}\approx 5\times {10}^{-8}$. (b) and (c) Taken in California on 3 May 2012.

Figure 6

A plot and photographs of a 3-in-2-out interaction, where there are three line solitons on one side of the interaction region and two line solitons on the other side. (a) $k_1=1$, $k_2=2$, $k_3=3$, $P_1=-1/3$, $P_2=-2/3$, $P_3=-5/3$. (b) and (c) Taken in California on 4 May 2012.