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

TL;DR

This study uses high-precision numerical simulations to investigate Fermi-Pasta-Ulam recurrence in water waves over finite depth, revealing conditions under which recurrence occurs and providing a formula for recurrence time.

Contribution

It demonstrates the occurrence of FPU recurrence in water waves over finite depth and derives a formula for the recurrence time based on initial conditions and physical parameters.

Findings

FPU recurrence occurs for moderate wave amplitudes and specific spatial periods.

Recurrence time scales with initial amplitude, wavelength, and depth as derived.

Numerical simulations confirm theoretical predictions of recurrence behavior.

Abstract

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 ${\bf v}=0$. 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 $T_{\rm FPU}$ is well fitted by the formula $T_{\rm FPU}(g/h)^{1/2}\approx 0.16(L/h)^2(h/A_0)^{1/2}$.