Nonlinear spatial evolution of degenerate quartets of water waves
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916155806973952 |
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| author | Heffernan, Conor Chabchoub, Amin Stuhlmeier, Raphael |
| author_facet | Heffernan, Conor Chabchoub, Amin Stuhlmeier, Raphael |
| contents | In this manuscript we investigate the Benjamin-Feir (or modulation) instability for the spatial evolution of water waves from the perspective of the discrete, spatial Zakharov equation, which captures cubically nonlinear and resonant wave interactions in deep water without restrictions on spectral bandwidth. Spatial evolution, with measurements at discrete locations, is pertinent for laboratory hydrodynamic experiments, such as in wave flumes, which rely on time-series measurements at a series of fixed gauges installed along the facility. This setting is likewise appropriate for experiments in electromagnetic and plasma waves. Through a reformulation of the problem for a degenerate quartet, we bring to bear techniques of phase-plane analysis which elucidate the full dynamics without recourse to linear stability analysis. In particular we find hitherto unexplored breather solutions and discuss the optimal transfer of energy from carrier to sidebands. Finally, we discuss the observability of such discrete solutions in light of numerical simulations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06558 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nonlinear spatial evolution of degenerate quartets of water waves Heffernan, Conor Chabchoub, Amin Stuhlmeier, Raphael Fluid Dynamics Dynamical Systems Pattern Formation and Solitons 76B15, 76E30, 70K30 In this manuscript we investigate the Benjamin-Feir (or modulation) instability for the spatial evolution of water waves from the perspective of the discrete, spatial Zakharov equation, which captures cubically nonlinear and resonant wave interactions in deep water without restrictions on spectral bandwidth. Spatial evolution, with measurements at discrete locations, is pertinent for laboratory hydrodynamic experiments, such as in wave flumes, which rely on time-series measurements at a series of fixed gauges installed along the facility. This setting is likewise appropriate for experiments in electromagnetic and plasma waves. Through a reformulation of the problem for a degenerate quartet, we bring to bear techniques of phase-plane analysis which elucidate the full dynamics without recourse to linear stability analysis. In particular we find hitherto unexplored breather solutions and discuss the optimal transfer of energy from carrier to sidebands. Finally, we discuss the observability of such discrete solutions in light of numerical simulations. |
| title | Nonlinear spatial evolution of degenerate quartets of water waves |
| topic | Fluid Dynamics Dynamical Systems Pattern Formation and Solitons 76B15, 76E30, 70K30 |
| url | https://arxiv.org/abs/2403.06558 |