论文标题
无限深度的空间周期性水波
Spatially quasi-periodic water waves of infinite depth
论文作者
论文摘要
我们在空间上的准周期环境中制定了二维重力水波方程，并介绍了初始值问题解决方案的数值研究。我们提出了一个运动方程的傅立叶伪 - 光谱离散化，其中一维准周期函数由圆环上的二维周期函数表示。我们采用保形映射公式，并采用希尔伯特变换的准周期版本来确定自由表面的正常速度。提出了两种时间stepping初始值问题的方法，即显式runge-kutta（ERK）方法和指数差异（ETD）方案。 ETD方法利用小规模的分解来消除由于表面张力引起的刚度。我们执行一项收敛研究，以比较行动波测试问题中方法的准确性和效率。我们还提供了一个周期性波轮廓的示例，其中包含垂直切线线，该线以准周期速度潜力启动。随着时间的流逝，每个波峰的发展都不同，只有一些波峰推翻。除了水波之外，我们认为空间准周期性是研究线性和非线性波的动力学的自然环境，为通常建模假设提供了第三种选择，即在周期域上进化或在无限限度衰减。
We formulate the two-dimensional gravity-capillary water wave equations in a spatially quasi-periodic setting and present a numerical study of solutions of the initial value problem. We propose a Fourier pseudo-spectral discretization of the equations of motion in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on a torus. We adopt a conformal mapping formulation and employ a quasi-periodic version of the Hilbert transform to determine the normal velocity of the free surface. Two methods of time-stepping the initial value problem are proposed, an explicit Runge-Kutta (ERK) method and an exponential time-differencing (ETD) scheme. The ETD approach makes use of the small-scale decomposition to eliminate stiffness due to surface tension. We perform a convergence study to compare the accuracy and efficiency of the methods on a traveling wave test problem. We also present an example of a periodic wave profile containing vertical tangent lines that is set in motion with a quasi-periodic velocity potential. As time evolves, each wave peak evolves differently, and only some of them overturn. Beyond water waves, we argue that spatial quasi-periodicity is a natural setting to study the dynamics of linear and nonlinear waves, offering a third option to the usual modeling assumption that solutions either evolve on a periodic domain or decay at infinity.
