  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.

Fluid Dynamics;Numerical Analysis;Numerical Analysis