Bifurcations of unstable eigenvalues for Stokes waves derived from conserved energy

2025/03/19 by Sergey A. Dyachenko, Dyachenko, Sergey, Dmitry E. Pelinovsky +1 · 1 citation
Earth and Planetary Sciences · Engineering · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Ocean Waves and Remote Sensing #Pattern Formation and Solitons (nlin.PS) #Wave and Wind Energy Systems

paper · pdf · doi:10.48550/arxiv.2503.15713

openalex publication_date 2025/03/19 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28

Abstract

We address Euler's equations for irrotational gravity waves in an infinitely deep fluid rewritten in conformal variables. Stokes waves are traveling waves with the smooth periodic profile. In agreement with the previous numerical results, we give a rigorous proof that the zero eigenvalue bifurcation in the linearized equations of motion for co-periodic perturbations occurs at each extremal point of the energy function versus the steepness parameter, provided that the wave speed is not extremal at the same steepness. We derive the normal form for the unstable eigenvalues and, assisted with numerical approximation of its coefficients, we show that the new unstable eigenvalues emerge only in the direction of increasing steepness.

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• Stokes Waves in Water of Finite Depth

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