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1. Spatially Quasi-Periodic Water Waves of Infinite Depth

   https://doi.org/10.1007/s00332-021-09689-2  

   Wilkening, Jon ; Zhao, Xinyu ( April 2021 , Journal of Nonlinear Science)

   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.

    

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2. Spatially quasi-periodic water waves of finite depth

   https://doi.org/10.1098/rspa.2023.0019  

   Wilkening, Jon ; Zhao, Xinyu ( April 2023 , Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   We present a numerical study of spatially quasi-periodic gravity-capillary waves of finite depth in both the initial value problem and travelling wave settings. We adopt a quasi-periodic conformal mapping formulation of the Euler equations, where one-dimensional quasi-periodic functions are represented by periodic functions on a higher-dimensional torus. We compute the time evolution of free surface waves in the presence of a background flow and a quasi-periodic bottom boundary and observe the formation of quasi-periodic patterns on the free surface. Two types of quasi-periodic travelling waves are computed: small-amplitude waves bifurcating from the zero-amplitude solution and larger-amplitude waves bifurcating from finite-amplitude periodic travelling waves. We derive weakly nonlinear approximations of the first type and investigate the associated small-divisor problem. We find that waves of the second type exhibit striking nonlinear behaviour, e.g. the peaks and troughs are shifted non-periodically from the corresponding periodic waves due to the activation of quasi-periodic modes.

    

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3. Nonlinear travelling internal waves with piecewise-linear shear profiles

   https://doi.org/10.1017/jfm.2018.679  

   Oliveras, K. L. ; Curtis, C. W. ( December 2018 , Journal of Fluid Mechanics)

   null (Ed.)

   In this work, we study the nonlinear travelling waves in density stratified fluids with piecewise-linear shear currents. Beginning with the formulation of the water-wave problem due to Ablowitz et al.  ( J. Fluid Mech. , vol. 562, 2006, pp. 313–343), we extend the work of Ashton & Fokas ( J. Fluid Mech. , vol. 689, 2011, pp. 129–148) and Haut & Ablowitz ( J. Fluid Mech. , vol. 631, 2009, pp. 375–396) to examine the interface between two fluids of differing densities and varying linear shear. We derive a systems of equations depending only on variables at the interface, and numerically solve for periodic travelling wave solutions using numerical continuation. Here, we consider only branches which bifurcate from solutions where there is no slip in the tangential velocity at the interface for the trivial flow. The spectral stability of these solutions is then determined using a numerical Fourier–Floquet technique. We find that the strength of the linear shear in each fluid impacts the stability of the corresponding travelling wave solutions. Specifically, opposing shears may amplify or suppress instabilities. 

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4. Spatially quasi-periodic bifurcations from periodic traveling water waves and a method for detecting bifurcations using signed singular values

   https://doi.org/10.1016/j.jcp.2023.111954  

   Wilkening, Jon ; Zhao, Xinyu ( April 2023 , Journal of Computational Physics)

   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 root-bracketing 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 quasi-periodic traveling water waves that bifurcate from large-amplitude periodic waves. The water wave equations are formulated in a conformal mapping framework to facilitate the computation of the quasi-periodic Dirichlet-Neumann operator. We find examples of pure gravity waves with zero surface tension and overhanging gravity-capillary waves. In both cases, the waves have two spatial quasi-periods 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 gravity-capillary wave problem demonstrates the effectiveness of using the signed smallest singular value as a test function for multi-parameter bifurcation problems. This test function becomes mesh independent once the mesh is fine enough. 

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5. Large-Amplitude Steady Solitary Water Waves with Constant Vorticity

   https://doi.org/10.1007/s00205-023-01841-4  

   Haziot, Susanna V. ; Wheeler, Miles H. ( March 2023 , Archive for Rational Mechanics and Analysis)

   This paper considers two-dimensional steady solitary waves with constant vorticity propagating under the influence of gravity over an impermeable flat bed. Unlike in previous works on solitary waves, we allow for both internal stagnation points and overhanging wave profiles. Using analytic global bifurcation theory, we construct continuous curves of large-amplitude solutions. Along these curves, either the wave amplitude approaches the maximum possible value, the dimensionless wave speed becomes unbounded, or a singularity develops in a conformal map describing the fluid domain. This is stronger than what one would expect from a straightforward generalization of existing results for periodic waves. We also show that an arbitrary solitary wave of elevation with constant vorticity must be supercritical. The existence proof relies on a novel reformulation of the problem as an elliptic system for two scalar functions in a fixed domain, one describing the conformal map of the fluid region and the other the flow beneath the wave.

    

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