We formulate the twodimensional gravitycapillary water wave equations in a spatially quasiperiodic setting and present a numerical study of solutions of the initial value problem. We propose a Fourier pseudospectral discretization of the equations of motion in which onedimensional quasiperiodic functions are represented by twodimensional periodic functions on a torus. We adopt a conformal mapping formulation and employ a quasiperiodic version of the Hilbert transform to determine the normal velocity of the free surface. Two methods of timestepping the initial value problem are proposed, an explicit Runge–Kutta (ERK) method and an exponential timedifferencing (ETD) scheme. The ETD approach makes use of the smallscale 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 quasiperiodic velocity potential. As time evolves, each wave peak evolves differently, and only some of them overturn. Beyond water waves, we argue that spatial quasiperiodicity 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.
more » « less Award ID(s):
 1716560
 NSFPAR ID:
 10377675
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Journal of Nonlinear Science
 Volume:
 31
 Issue:
 3
 ISSN:
 09388974
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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