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1. 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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2. Controlling Flow Separation on a Thick Airfoil Using Backward Traveling Waves

   https://doi.org/10.2514/1.J059428  

   Akbarzadeh, A. M.; Borazjani, I. (June 2020, AIAA Journal)

   Large-eddy simulations (LESs) of low-Reynolds-number flow (Re=50,000) over a NACA0018 airfoil are performed to investigate flow control at the stall angle of attack (15 deg) by low-amplitude surface waves (actuations) of different types (backward/forward traveling and standing waves) on the airfoil’s suction side. It is found that the backward (toward downstream) traveling waves, inspired from aquatic swimmers, are more effective than forward traveling and standing wave actuations. The results of simulations show that a backward traveling wave with a reduced frequency f∗=4 (f∗=fL/U, where f is frequency; L, chord length; and U, free flow velocity), a nondimensional wavelength λ∗=0.2 (λ∗=λ/L, where λ is dimensional wavelength), and a nondimensional amplitude a∗=0.002 (a∗=a/L, where a is dimensional amplitude) can suppress stall. In contrast, the flow over the airfoil with either standing or forward traveling wave actuations separates from the leading edge similar to the baseline. Consequently, the backward traveling wave creates the highest lift-to-drag ratio. For traveling waves at a higher amplitude (a∗=0.008), however, the shear layer becomes unstable from the actuation point and creates periodic coherent structures. Therefore, the lift coefficient decreases compared with the low-amplitude case. 

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3. Whitham modulation theory for the Zakharov–Kuznetsov equation and stability analysis of its periodic traveling wave solutions

   https://doi.org/10.1111/sapm.12651  

   Biondini, Gino; Chernyavsky, Alexander (February 2024, Studies in Applied Mathematics)

   Abstract We derive the Whitham modulation equations for the Zakharov–Kuznetsov equation via a multiple scales expansion and averaging two conservation laws over one oscillation period of its periodic traveling wave solutions. We then use the Whitham modulation equations to study the transverse stability of the periodic traveling wave solutions. We find that all periodic solutions traveling along the first spatial coordinate are linearly unstable with respect to purely transversal perturbations, and we obtain an explicit expression for the growth rate of perturbations in the long wave limit. We validate these predictions by linearizing the equation around its periodic solutions and solving the resulting eigenvalue problem numerically. We also calculate the growth rate of the solitary waves analytically. The predictions of Whitham modulation theory are in excellent agreement with both of these approaches. Finally, we generalize the stability analysis to periodic waves traveling in arbitrary directions and to perturbations that are not purely transversal, and we determine the resulting domains of stability and instability. 

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4. Quasi-periodic travelling gravity–capillary waves

   https://doi.org/10.1017/jfm.2021.28  

   Wilkening, Jon; Zhao, Xinyu (May 2021, Journal of Fluid Mechanics)

   We present a numerical study of spatially quasi-periodic 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 quasi-periodic 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 pseudo-spectral discretization of the travelling water wave equations in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on the torus. This leads to an overdetermined nonlinear least-squares problem that we solve using a variant of the Levenberg–Marquardt method. We investigate various properties of quasi-periodic 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 two-parameter family of waves. 

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5. Standing and traveling waves in a minimal nonlinearly dispersive lattice model

   Parker, R; Germain, P; Cuevas_Maraver, J; Aceves, A; Kevrekidis, PG (June 2024, Physica D Nonlinear phenomena)

   In the work of Colliander et al. (2020) a minimal lattice model was constructed describing the transfer of energy to high frequencies in the defocusing nonlinear Schrödinger equation. In the present work, we present a systematic study of the coherent structures, both standing and traveling, that arise in the context of this model. We find that the nonlinearly dispersive nature of the model is responsible for standing waves in the form of discrete compactons. On the other hand, analysis of the dynamical features of the simplest nontrivial variant of the model, namely the dimer case, yields both solutions where the intensity is trapped in a single site and solutions where the intensity moves between the two sites, which suggests the possibility of moving excitations in larger lattices. Such excitations are also suggested by the dynamical evolution associated with modulational instability. Our numerical computations confirm this expectation, and we systematically construct such traveling states as exact solutions in lattices of varying size, as well as explore their stability. A remarkable feature of these traveling lattice waves is that they are of ‘‘antidark’’ type, i.e., they are mounted on top of a non-vanishing background. These studies shed light on the existence, stability and dynamics of such standing and traveling states in 1 + 1 dimensions, and pave the way for exploration of corresponding configurations in higher dimensions. 

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