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Title: Spatially quasi-periodic water waves of finite depth

We present a numerical study of spatially quasi-periodic gravity-capillary waves of finite depth in both the initial value problem and travelling 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 travelling waves are computed: small-amplitude waves bifurcating from the zero-amplitude solution and larger-amplitude waves bifurcating from finite-amplitude periodic travelling 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 behaviour, e.g. the peaks and troughs are shifted non-periodically from the corresponding periodic waves due to the activation of quasi-periodic modes.

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Award ID(s):
Author(s) / Creator(s):
Publisher / Repository:
Royal Society Publishing
Date Published:
Journal Name:
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Page Range / eLocation ID:
Subject(s) / Keyword(s):
["gravity-capillary waves, bottom topography, conformal map, bifurcation"]
Medium: X
Sponsoring Org:
National Science Foundation
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