First isola of modulational instability of Stokes waves in deep water

2024/01/26 by Berti, Massimiliano, Maspero, Alberto, Ventura, Paolo · 3 citations
#35B35 #76B15 (Primary) #76E99 (Secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2401.14689

Abstract

We prove high-frequency modulational instability of small-amplitude Stokes waves in deep water under longitudinal perturbations, providing the first isola of unstable eigenvalues branching off from \mathtti\frac34. Unlike the finite depth case this is a degenerate problem and the real part of the unstable eigenvalues has a much smaller size than in finite depth. By a symplectic version of Kato theory we reduce to search the eigenvalues of a 2× 2 Hamiltonian and reversible matrix which has eigenvalues with non-zero real part if and only if a certain analytic function is not identically zero. In deep water we prove that the Taylor coefficients up to order three of this function vanish, but not the fourth-order one.

Cited by

• Long time dynamics of space periodic water waves
• Full Benjamin-Feir instability of capillary-gravity Stokes waves in finite depth
• Instabilities of internal gravity waves in the two-dimensional Boussinesq system

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• Proof of modulational instability of Stokes waves in deep water
• Stokes waves at the critical depth are modulational unstable
• Infinitely many isolas of modulational instability for Stokes waves
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• Benjamin-Feir instability of Stokes waves in finite depth