Spatially quasi-periodic water waves of finite depth
Spatially quasi-periodic water waves of finite depth
复制标题
DOI：
10.1098/rspa.2023.0019
复制
发表时间：
2023-01
期刊：
影响因子：
--
通讯作者：
J. Wilkening;Xinyu Zhao
中科院分区：
文献类型：
--
作者：
J. Wilkening;Xinyu Zhao
我们在初始值问题和行波设置中对有限深度的空间准周期重力毛细管波进行了数值研究。我们采用欧拉方程的准周期共形映射公式，其中一维准周期函数由高维环面上的周期函数表示。我们计算了存在背景流和准周期底部边界的情况下自由表面波的时间演化，并观察了自由表面上准周期模式的形成。计算两种类型的准周期行波：从零幅度解分叉的小幅度波和从有限幅度周期性行波分叉的较大幅度波。我们推导了第一类的弱非线性近似并研究了相关的小除数问题。我们发现第二类波表现出惊人的非线性行为，例如由于准周期模式的激活，波峰和波谷相对于相应的周期波发生非周期性移动。
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.