An integrable evolution equation for surface waves in deep water

2011/01/30 by R. A. Kraenkel, Kraenkel, R., Hervé Leblond +3
Earth and Planetary Sciences · Mathematics · Physics and Astronomy · #Differential Equations and Numerical Methods #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Ocean Waves and Remote Sensing

paper · doi:10.48550/arxiv.1101.5773

openalex publication_date 2011/01/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In order to describe the dynamics of monochromatic surface waves in deep water, we derive a nonlinear and dispersive system of equations for the free surface elevation and the free surface velocity from the Euler equations in infinite depth. From it, and using a multiscale perturbative methods, an asymptotic model for small-aspect-ratio waves is derived. The model is shown to be completely integrable. The Lax pair, the first conserved quantities as well as the symmetries are exhibited. Theoretical and numerical studies reveal that it supports periodic progressive Stokes waves which peak and break in finite time. Comparison between the limiting wave solution of the asymptotic model and classical irrotational results is performed.

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