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 ₀cos(2πx/L), and the velocity field 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 ≲ 0.12h and spatial periods at least L ≲ 120h. The time of recurrence T _FPU is well fitted by the formula T _FPU(g/h)^1/2 ≈ 0.16(L/h)²(h/A ₀)^1/2.