A high-order asymptotic analysis of the Benjamin-Feir instability spectrum in arbitrary depth

Article Mechanics

High-frequency instabilities of Stokes waves

Ryan P. Creedon et al.

Summary: This study investigates the spectral stability of small-amplitude, one-dimensional Stokes waves on the surface of an incompressible, inviscid and irrotational fluid. By using a perturbation method, the first few high-frequency instabilities of the small-amplitude Stokes waves are accurately described and compared through asymptotic and numerical computations.

JOURNAL OF FLUID MECHANICS (2022)

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Article Mathematics

Full description of Benjamin-Feir instability of stokes waves in deep water

Massimiliano Berti et al.

Summary: In this study, we investigate the linear instability of small-amplitude, traveling, space periodic solutions (Stokes waves) of the 2D gravity water waves equations in deep water with respect to long-wave perturbations. We describe the behavior of the four eigenvalues close to zero of the linearized equations at the Stokes wave and prove that a pair of non-purely imaginary eigenvalues forms a closed figure 8, which is consistent with numerical simulations. Our approach combines spectral methods and similarity transformation theory.

INVENTIONES MATHEMATICAE (2022)

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Article Physics, Fluids & Plasmas

High-Frequency Instabilities of a Boussinesq-Whitham System: A Perturbative Approach

Ryan Creedon et al.

Summary: The spectral stability of small-amplitude, periodic, traveling-wave solutions in a Boussinesq-Whitham system is analyzed. Numerical simulations show that these solutions exhibit high-frequency instabilities when subjected to bounded perturbations on the real line. A formal perturbation method is used to estimate the asymptotic behavior of these instabilities in the small-amplitude regime, and the results are compared with direct numerical computations.

FLUIDS (2021)

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Article Mathematics, Applied

High-Frequency Instabilities of the Kawahara Equation: A Perturbative Approach

Ryan Creedon et al.

Summary: In this study, the spectral stability of small-amplitude, periodic, traveling-wave solutions of the Kawahara equation is analyzed, revealing high-frequency instabilities when subject to bounded perturbations on the whole real line. A formal perturbation method is introduced to determine the asymptotic growth rates of these instabilities and other properties. Explicit numerical computations are utilized to confirm the asymptotic results.

SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS (2021)

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Article Mathematics, Applied