UC Berkeley

In this thesis, we study the problem of two dimensional spatially quasi-periodic gravity-capillary waves on the surface of an ideal fluid of infinite depth. We formulate the water wave equations in a spatially quasi-periodic setting and present a numerical study of solutions of both the initial value problem and the traveling wave 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 time-stepping schemes of the initial value problem are proposed, an explicit Runge-Kutta (ERK) method and an exponential time-differencing (ETD) scheme. We present an example of a periodic wave profile containing vertical tangent lines that is set in motion with a quasi-periodic velocity potential. We formulate the traveling wave problem as a nonlinear least squares problem that we solve using a variant of the Levenberg-Marquardt method. Two types of quasi-periodic traveling solutions are computed: small-amplitude solutions that bifurcate from the zero solution and large-amplitude solutions that bifurcate from finite-amplitude periodic traveling solutions. Solutions of the first type are identified by two bifurcation parameters. We also compute the leading terms of the asymptotic expansion of the solution using these parameters. For solutions of the second type, we apply the Fourier-Bloch decomposition to study the linearization around periodic traveling solutions and obtain a one-parameter family of quasi-periodic solutions bifurcating from the branch of periodic solutions. As an example, we compute a branch of quasi-periodic overturning traveling solutions that bifurcate from a periodic overturning traveling solution.