Spatially quasi-periodic water waves of finite depth

TL;DR

This paper numerically investigates finite-depth gravity-capillary water waves with quasi-periodic patterns, revealing complex nonlinear behaviors and bifurcations in both initial value and traveling wave scenarios.

Contribution

It introduces a quasi-periodic conformal mapping approach to study quasi-periodic water waves, including bifurcation analysis and nonlinear behavior characterization.

Findings

Observation of quasi-periodic pattern formation on free surfaces.

Identification of bifurcation types for traveling waves.

Discovery of non-periodic peak and trough shifts in nonlinear waves.

Abstract

We present a numerical study of spatially quasi-periodic gravity-capillary waves of finite depth in both the initial value problem and traveling wave settings. We adopt a quasi-periodic conformal mapping formulation of the Euler equations, where one-dimensional quasi-periodic functions are represented by periodic functions on a higher-dimensional torus. We compute the time evolution of free surface waves in the presence of a background flow and a quasi-periodic bottom boundary and observe the formation of quasi-periodic patterns on the free surface. Two types of quasi-periodic traveling waves are computed: small-amplitude waves bifurcating from the zero-amplitude solution and larger-amplitude waves bifurcating from finite-amplitude periodic traveling waves. We derive weakly nonlinear approximations of the first type and investigate the associated small-divisor problem. We find that waves of the second type exhibit striking nonlinear behavior, e.g., the peaks and troughs are shifted non-periodically from the corresponding periodic waves due to the activation of quasi-periodic modes.