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arXiv · 2208.08172

The nonlinear Benjamin-Feir instability -- Hamiltonian dynamics, primitive breathers, and steady solutions

David Andrade·Raphael Stuhlmeier
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Abstract

We develop a general framework to describe the cubically nonlinear interaction of a unidirectional degenerate quartet of deep-water gravity waves. Starting from the discretised Zakharov equation, and thus without restriction on spectral bandwidth, we derive a planar Hamiltonian system in terms of the dynamic phase and a modal amplitude. This is characterised by two free parameters: the wave action and the mode separation between the carrier and the side-bands. The mode separation serves as a bifurcation parameter, which allows us to fully classify the dynamics. Centres of our system correspond to non-trivial, steady-state nearly-resonant degenerate quartets. The existence of saddle-points is connected to the instability of uniform and bichromatic wave trains, generalising the classical picture of the Benjamin-Feir instability. Moreover, heteroclinic orbits are found to correspond to primitive, three-mode breather solutions, including an analogue of the famed Akhmediev breather solution of the nonlinear Schr\"odinger equation.

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David Andrade, Raphael Stuhlmeier. 2022-08-17. The nonlinear Benjamin-Feir instability -- Hamiltonian dynamics, primitive breathers, and steady solutions. https://doi.org/10.1017/jfm.2023.96

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