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Modulational Instability in the Whitham Equation for Water Waves
Author(s) -
Hur Vera Mikyoung,
Johnson Mathew A.
Publication year - 2015
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12061
Subject(s) - modulational instability , amplitude , floquet theory , dispersion relation , instability , nonlinear system , mathematical analysis , waves and shallow water , physics , wavelength , perturbation (astronomy) , mathematics , integrable system , stokes wave , classical mechanics , breaking wave , wave propagation , mechanics , optics , quantum mechanics , thermodynamics
We show that periodic traveling waves with sufficiently small amplitudes of the Whitham equation, which incorporates the dispersion relation of surface water waves and the nonlinearity of the shallow water equations, are spectrally unstable to long‐wavelengths perturbations if the wave number is greater than a critical value, bearing out the Benjamin–Feir instability of Stokes waves; they are spectrally stable to square integrable perturbations otherwise. The proof involves a spectral perturbation of the associated linearized operator with respect to the Floquet exponent and the small‐amplitude parameter. We extend the result to related, nonlinear dispersive equations.