Geodesic
First isola of modulational instability of Stokes waves in deep water
Massimiliano Berti1 orcid ; Alberto Maspero1 orcid ; Paolo Ventura2 orcid
1Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy
2Università degli Studi di Milano, Milano, Italy
EMS surveys in mathematical sciences, Tome 12 (2025) no. 1, pp. 79-122

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DOI

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 i43​. 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 nonzero 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.
DOI : 10.4171/emss/91
Classification : 76B15, 47A55
Mots-clés : modulational instability, Stokes waves, water waves, perturbation of eigenvalues

Massimiliano Berti  1   ; Alberto Maspero  1   ; Paolo Ventura  2

1 Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy
2 Università degli Studi di Milano, Milano, Italy 
Massimiliano Berti; Alberto Maspero; Paolo Ventura. First isola of modulational instability of Stokes waves in deep water. EMS surveys in mathematical sciences, Tome 12 (2025) no. 1, pp. 79-122. doi: 10.4171/emss/91
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     year = {2025},
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     url = {http://geodesic.mathdoc.fr/articles/10.4171/emss/91/}
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