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.

我们在空间准周期的环境中建立了二维重力-毛细水波方程，并给出了初值问题解的数值研究。我们提出运动方程的傅立叶拟谱离散化，其中一维拟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法向速度。提出了两种对初始值问题进行时间分步的方法，即显式Runge-Kutta（ERK）方法和指数时差（ETD）方法。ETD方法利用小规模分解来消除由于表面张力引起的刚度。我们进行了收敛研究，比较了行波测试问题上方法的准确性和效率。我们还提供了一个包含垂直切线的周期波轮廓的示例，该切线以准周期速度势运动。随着时间的推移，每个波峰的演变都不同，只有其中一些会翻转。除了水波，我们认为空间准周期是研究线性和非线性波动力学的自然环境，为通常的建模假设（解决方案在周期域上演化或在无穷大处衰减）提供了第三种选择。每个波峰的演变都不同，只有其中一些会倾覆。除了水波，我们认为空间准周期是研究线性和非线性波动力学的自然环境，为通常的建模假设（解决方案在周期域上演化或在无穷大处衰减）提供了第三种选择。每个波峰的演变都不同，只有其中一些会倾覆。除了水波，我们认为空间准周期是研究线性和非线性波动力学的自然环境，为通常的建模假设（解决方案在周期域上演化或在无穷大处衰减）提供了第三种选择。