Numerical study of Fermi-Pasta-Ulam recurrence for water waves over finite depth

Highly accurate direct numerical simulations have been performed for two-dimensional free-surface potential flows of an ideal incompressible fluid over a constant depth $h$, in the gravity field $g$. In each numerical experiment, at $t=0$ the free surface profile was in the form $y=A_0\cos(2\pi x/L)$, and the velocity field $\mathbf v=0$. The computations demonstrate the phenomenon of Fermi-Pasta-Ulam (FPU) recurrence takes place in such systems for moderate initial wave amplitudes $A_0\lesssim 0.12 h$ and spatial periods at least $L\lesssim 120 h$. The time of recurrence $T_{\mathrm{FPU}}$ is well fitted by the formula $T_{\mathrm{FPU}}(g/h)^{1/2}\approx 0.16(L/h)^2(h/A_0)^{1/2}$.