Spatially quasi-periodic gravity-capillary water waves of infinite depth
We formulate the two-dimensional gravity-capillary water wave equations in a spatially quasi-periodic setting and present a numerical study of traveling waves and more general solutions of the initial value problem. The former are a generalization of the classical Wilton ripple problem. 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. We compute traveling waves in a nonlinear least-squares framework using a variant of the Levenberg-Marquardt method. We propose four methods for timestepping the initial value problem, two explicit Runge-Kutta (ERK) methods and two exponential time-differencing (ETD) schemes. The latter approach makes use of the small-scale decomposition to eliminate stiffness due to surface tension. We investigate various properties of quasi-periodic traveling waves, including Fourier resonances and the dependence of wave speed and surface tension on the amplitude parameters that describe a two-parameter family of waves. 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 that causes some of the waves to overturn and others to flatten out as time evolves.
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