ORCID
- Stuhlmeier, Raphael: 0000-0002-6568-1543
Abstract
We develop a general framework to describe the cubically nonlinear interaction of a degenerate quartet of deep-water gravity waves in one or two spatial dimensions. 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 sidebands. For unidirectional waves, 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 discrete, three-mode breather solutions, including an analogue of the famed Akhmediev breather solution of the nonlinear Schrödinger equation.
DOI
10.1017/jfm.2023.96
Publication Date
2023-03-10
Publication Title
Journal of Fluid Mechanics
Volume
958
ISSN
0022-1120
Embargo Period
2023-03-21
Organisational Unit
School of Engineering, Computing and Mathematics
Recommended Citation
Andrade, D., & Stuhlmeier, R. (2023) 'The nonlinear Benjamin–Feir instability – Hamiltonian dynamics, discrete breathers and steady solutions', Journal of Fluid Mechanics, 958. Available at: https://doi.org/10.1017/jfm.2023.96
