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We present a method of detecting bifurcations by locating zeros of a signed version of the smallest singular value of the Jacobian. This enables the use of quadratically convergent rootbracketing techniques or Chebyshev interpolation to locate bifurcation points. Only positive singular values have to be computed, though the method relies on the existence of an analytic or smooth singular value decomposition (SVD). The sign of the determinant of the Jacobian, computed as part of the bidiagonal reduction in the SVD algorithm, eliminates slope discontinuities at the zeros of the smallest singular value. We use the method to search for spatially quasiperiodic traveling water waves that bifurcate from largeamplitude periodic waves. The water wave equations are formulated in a conformal mapping framework to facilitate the computation of the quasiperiodic DirichletNeumann operator. We find examples of pure gravity waves with zero surface tension and overhanging gravitycapillary waves. In both cases, the waves have two spatial quasiperiods whose ratio is irrational. We follow the secondary branches via numerical continuation beyond the realm of linearization about solutions on the primary branch to obtain traveling water waves that extend over the real line with no two crests or troughs of exactly the same shape. The pure gravity wave problem is of relevance to ocean waves, where capillary effects can be neglected. Such waves can only exist through secondary bifurcation as they do not persist to zero amplitude. The gravitycapillary wave problem demonstrates the effectiveness of using the signed smallest singular value as a test function for multiparameter bifurcation problems. This test function becomes mesh independent once the mesh is fine enough.more » « less

We present a numerical study of spatially quasiperiodic gravitycapillary waves of finite depth in both the initial value problem and travelling wave settings. We adopt a quasiperiodic conformal mapping formulation of the Euler equations, where onedimensional quasiperiodic functions are represented by periodic functions on a higherdimensional torus. We compute the time evolution of free surface waves in the presence of a background flow and a quasiperiodic bottom boundary and observe the formation of quasiperiodic patterns on the free surface. Two types of quasiperiodic travelling waves are computed: smallamplitude waves bifurcating from the zeroamplitude solution and largeramplitude waves bifurcating from finiteamplitude periodic travelling waves. We derive weakly nonlinear approximations of the first type and investigate the associated smalldivisor problem. We find that waves of the second type exhibit striking nonlinear behaviour, e.g. the peaks and troughs are shifted nonperiodically from the corresponding periodic waves due to the activation of quasiperiodic modes.

Abstract We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface, are compatible with arbitrary parameterizations of the free surface and boundaries, and allow for circulation around each obstacle, which leads to multiplevalued velocity potentials but singlevalued stream functions. We prove that the resulting secondkind Fredholm integral equations are invertible, possibly after a physically motivated finiterank correction. In an anglearclength setting, we show how to avoid curve reconstruction errors that are incompatible with spatial periodicity. We use the proposed methods to study gravitycapillary waves generated by flow around several elliptical obstacles above a flat or variable bottom boundary. In each case, the free surface eventually selfintersects in a splash singularity or collides with a boundary. We also show how to evaluate the velocity and pressure with spectral accuracy throughout the fluid, including near the free surface and solid boundaries. To assess the accuracy of the time evolution, we monitor energy conservation and the decay of Fourier modes and compare the numerical results of the two methods to each other. We implement several solvers for the discretized linear systems and compare their performance. The fastest approach employs a graphics processing unit (GPU) to construct the matrices and carry out iterations of the generalized minimal residual method (GMRES).

We propose a new twoparameter family of hybrid travelingstanding (TS) water waves in infinite depth that evolve to a spatial translation of their initial condition at a later time. We use the square root of the energy as an amplitude parameter and introduce a traveling parameter that naturally interpolates between pure traveling waves moving in either direction and pure standing waves in one of four natural phase configurations. The problem is formulated as a twopoint boundary value problem and a quasiperiodic torus representation is presented that exhibits TSwaves as nonlinear superpositions of counterpropagating traveling waves. We use an overdetermined shooting method to compute nearly 50,000 TSwave solutions and explore their properties. Examples of waves that periodically form sharp crests with high curvature or dimpled crests with negative curvature are presented. We find that pure traveling waves maximize the magnitude of the horizontal momentum among TSwaves of a given energy. Numerical evidence suggests that the twoparameter family of TSwaves contains many gaps and disconnections where solutions with the given parameters do not exist. Some of these gaps are shown to persist to zeroamplitude in a fourthorder perturbation expansion of the solutions in powers of the amplitude parameter. Analytic formulas for the coefficients of this perturbation expansion are identified using Chebyshev interpolation of solutions computed in quadrupleprecision.

We present a numerical study of spatially quasiperiodic travelling waves on the surface of an ideal fluid of infinite depth. This is a generalization of the classic Wilton ripple problem to the case when the ratio of wavenumbers satisfying the dispersion relation is irrational. We propose a conformal mapping formulation of the water wave equations that employs a quasiperiodic variant of the Hilbert transform to compute the normal velocity of the fluid from its velocity potential on the free surface. We develop a Fourier pseudospectral discretization of the travelling water wave equations in which onedimensional quasiperiodic functions are represented by twodimensional periodic functions on the torus. This leads to an overdetermined nonlinear leastsquares problem that we solve using a variant of the Levenberg–Marquardt method. We investigate various properties of quasiperiodic travelling waves, including Fourier resonances, time evolution in conformal space on the torus, asymmetric wave crests, capillary wave patterns that change from one gravity wave trough to the next without repeating and the dependence of wave speed and surface tension on the amplitude parameters that describe a twoparameter family of waves.more » « less

Abstract 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.

In this note we prove several analytical results about generalized Kimura diffusion operators, $L,$ defined on compact manifolds with corners, $P.$ It is shown that the $\cC^0(P)$graph closure of $L$ acting on $\cC^2(P)$ always has a compact resolvent. In the $1d$case, where $P=[0,1],$ we also establish a gradient estimate $\\pa_x f\_{\cC^0([0,1])}\leq C\ L f\_{\cC^0([0,1])},$ provided that $L$ has strictly positive weights at $\pa [0,1]=\{0,1\}.$ This in turn leads to a precise characterization of the domain of the $\cC^0$graph closure in this case.more » « less