The Fermi-Pasta-Ulam recurrence and related phenomena for 1D shallow-water waves in a finite basin

TL;DR

This study numerically investigates Fermi-Pasta-Ulam recurrence phenomena in fully nonlinear 1D shallow-water waves within a finite basin, revealing complex behaviors related to soliton dynamics and bed profile effects.

Contribution

It provides a detailed numerical analysis of FPU recurrence in nonlinear water waves, linking it to soliton interactions and bed profile variations, with analytical solutions for soliton configurations.

Findings

FPU recurrence varies with soliton number and parameters.

High and rogue waves emerge depending on bed profile and initial conditions.

Complex recurrence patterns can be simplified or absent based on system parameters.

Abstract

In this work, different regimes of the Fermi-Pasta-Ulam (FPU) recurrence are simulated numerically for fully nonlinear "one-dimensional" potential water waves in a finite-depth flume between two vertical walls. In such systems, the FPU recurrence is closely related to the dynamics of coherent structures approximately corresponding to solitons of the integrable Boussinesq system. A simplest periodic solution of the Boussinesq model, describing a single soliton between the walls, is presented in an analytical form in terms of the elliptic Jacobi functions. In the numerical experiments, it is observed that depending on a number of solitons in the flume and their parameters, the FPU recurrence can occur in a simple or complicated manner, or be practically absent. For comparison, the nonlinear dynamics of potential water waves over nonuniform beds is simulated, with initial states taken in the form of several pairs of colliding solitons. With a mild-slope bed profile, a typical phenomenon in the course of evolution is appearance of relatively high (rogue) waves, while for random, relatively short-correlated bed profiles it is either appearance of tall waves, or formation of sharp crests at moderate-height waves.