Modulational Instability in the Whitham Equation for Water Waves

DOI10.1111/sapm.12061zbMath1309.35079arXiv1312.1579OpenAlexW2963661406MaRDI QIDQ5175784

Mathew A. Johnson, Vera Mikyoung Hur

Publication date: 25 February 2015

Published in: Studies in Applied Mathematics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1312.1579

zbMATH Keywords

Stokes wavesperiodic traveling wavesWhitham equation

Mathematics Subject Classification ID

PDEs in connection with fluid mechanics (35Q35) Periodic solutions to PDEs (35B10) Nonlinear effects in hydrodynamic stability (76E30) Water waves, gravity waves; dispersion and scattering, nonlinear interaction (76B15) Waves for incompressible viscous fluids (76D33) Traveling wave solutions (35C07)

Related Items (44)

Modulational instability in nonlinear nonlocal equations of regularized long wave type ⋮ The cubic vortical Whitham equation ⋮ Shallow water models with constant vorticity ⋮ High-frequency instabilities of the Ostrovsky equation ⋮ Long time well-posedness of Whitham–Boussinesq systems ⋮ A direct construction of a full family of Whitham solitary waves ⋮ Modulational instability in the full-dispersion Camassa–Holm equation ⋮ The Whitham equation as a model for surface water waves ⋮ The Whitham equation with surface tension ⋮ Stability of Periodic Traveling Waves for Nonlinear Dispersive Equations ⋮ Modulational instability in the Whitham equation with surface tension and vorticity ⋮ Proof of Modulational Instability of Stokes Waves in Deep Water ⋮ Benjamin-Feir instability of Stokes waves in finite depth ⋮ Solitary waves in a Whitham equation with small surface tension ⋮ Bidirectional Whitham equations as models of waves on shallow water ⋮ Transverse spectral instabilities in Konopelchenko–Dubrovsky equation ⋮ Numerical bifurcation for the capillary Whitham equation ⋮ Unstable Stokes waves ⋮ Stokes waves at the critical depth are modulationally unstable ⋮ A numerical study of the Whitham equation as a model for steady surface water waves ⋮ Wave Breaking in a Shallow Water Model ⋮ Nonlinear modulational instability of dispersive PDE models ⋮ Existence of a highest wave in a fully dispersive two-way shallow water model ⋮ Stability of periodic progressive gravity wave solutions of the Whitham equation in the presence of vorticity ⋮ Hyperbolicity of the modulation equations for the Serre-Green-Naghdi model ⋮ A Method for Identifying Stability Regimes Using Roots of a Reduced-Order Polynomial ⋮ Symmetry and decay of traveling wave solutions to the Whitham equation ⋮ Numerical bifurcation and stability for the capillary-gravity Whitham equation ⋮ Stability of viscous St. Venant roll waves: from onset to infinite Froude number limit ⋮ High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs ⋮ A comparative study of bi-directional Whitham systems ⋮ The effects of surface tension on modulational instability in full-dispersion water-wave models ⋮ Wave breaking in the Whitham equation ⋮ Stability of traveling wave solutions to the Whitham equation ⋮ Modulational instability in a full‐dispersion shallow water model ⋮ Solitary waves in dispersive evolution equations of Whitham type with nonlinearities of mild regularity ⋮ Fully dispersive models for moving loads on ice sheets ⋮ Convergence of Petviashvili's Method near Periodic Waves in the Fractional Korteweg--de Vries Equation ⋮ Wilton ripples in weakly nonlinear models of water waves: existence and computation ⋮ On Whitham's conjecture of a highest cusped wave for a nonlocal dispersive equation ⋮ Full description of Benjamin-Feir instability of Stokes waves in deep water ⋮ A numerical approach to blow-up issues for dispersive perturbations of Burgers' equation ⋮ Modulational instabilities of periodic traveling waves in deep water ⋮ Existence of solitary-wave solutions to nonlocal equations

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