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| Main Author: | |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.24997 |
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| _version_ | 1866910252495011840 |
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| author | Simson, Päivo |
| author_facet | Simson, Päivo |
| contents | A unidirectional reduction of the deep-water surface gravity wave problem is derived in physical space using real variables. By employing a near-identity canonical transformation, cubic interactions are eliminated from the Hamiltonian, with an exact elimination of second- and third-order bound waves. A projection operator is then constructed to isolate the unidirectional, rightward-propagating dynamics at the next asymptotic order, yielding a single nonlocal evolution equation. The model admits the third-order Stokes wave as an exact monochromatic solution, and a multiple-scales analysis recovers the Dysthe envelope equation, including the nonlocal mean-flow coupling, without requiring an auxiliary boundary value problem. Dropping four sub-leading nonlinear terms that vanish on the resonant manifold yields a more compact variant suitable for analytical study. Numerical validations demonstrate that both formulations faithfully reproduce the full Euler dynamics through modulational-instability recurrence and broadband focusing up to moderate wave steepness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_24997 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A real-variable unidirectional reduction of deep-water gravity waves Simson, Päivo Fluid Dynamics A unidirectional reduction of the deep-water surface gravity wave problem is derived in physical space using real variables. By employing a near-identity canonical transformation, cubic interactions are eliminated from the Hamiltonian, with an exact elimination of second- and third-order bound waves. A projection operator is then constructed to isolate the unidirectional, rightward-propagating dynamics at the next asymptotic order, yielding a single nonlocal evolution equation. The model admits the third-order Stokes wave as an exact monochromatic solution, and a multiple-scales analysis recovers the Dysthe envelope equation, including the nonlocal mean-flow coupling, without requiring an auxiliary boundary value problem. Dropping four sub-leading nonlinear terms that vanish on the resonant manifold yields a more compact variant suitable for analytical study. Numerical validations demonstrate that both formulations faithfully reproduce the full Euler dynamics through modulational-instability recurrence and broadband focusing up to moderate wave steepness. |
| title | A real-variable unidirectional reduction of deep-water gravity waves |
| topic | Fluid Dynamics |
| url | https://arxiv.org/abs/2605.24997 |