A real-variable unidirectional reduction of deep-water gravity waves
Abstract
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, resulting in a bidirectional system analogous to the 1D Zakharov equation. An exact factorization of the linear dynamics then selects the rightward-propagating branch, which is carried to second order in the wave steepness, yielding a single nonlocal evolution equation with full linear dispersion. The model admits an exact monochromatic solution whose reconstructed surface elevation is the third-order Stokes wave, and a multiple-scales analysis recovers the Dysthe envelope equation complete with its nonlocal mean-flow term. The equation also admits exact breather solutions, whose profiles agree closely with those of the super compact equation. Dropping the sub-leading nonlinear terms that vanish on the resonant manifold provides a more compact variant suitable for analytical study. Numerical comparisons show that both formulations track the fully nonlinear Euler dynamics closely through modulational-instability recurrence and broadband focusing up to moderate wave steepness.
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BibTeX
@misc{simson2026realvariable,
title = {{A real-variable unidirectional reduction of deep-water gravity waves}},
author = {Päivo Simson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.24997},
eprint = {2605.24997},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.24997},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A real-variable unidirectional reduction of deep-water gravity waves
AU - Päivo Simson
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2605.24997
UR - https://arxiv.org/abs/2605.24997
AB - 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, resulting in a bidirectional system analogous to the 1D Zakharov equation. An exact factorization of the linear dynamics then selects the rightward-propagating branch, which is carried to second order in the wave steepness, yielding a single nonlocal evolution equation with full linear dispersion. The model admits an exact monochromatic solution whose reconstructed surface elevation is the third-order Stokes wave, and a multiple-scales analysis recovers the Dysthe envelope equation complete with its nonlocal mean-flow term. The equation also admits exact breather solutions, whose profiles agree closely with those of the super compact equation. Dropping the sub-leading nonlinear terms that vanish on the resonant manifold provides a more compact variant suitable for analytical study. Numerical comparisons show that both formulations track the fully nonlinear Euler dynamics closely through modulational-instability recurrence and broadband focusing up to moderate wave steepness.
ER -