Six-wave scattering in Hamiltonian Dysthe equations: Euler matching and integrability obstructions
Abstract
We compute the effective six-wave interaction in spatial and temporal Hamiltonian Dysthe equations and compare it with the complete degree-six coefficient of the deep-water Euler Hamiltonian. On every compact subset K of a generic quadratic-dispersion resonance patch, exact resonances persist under a small cubic correction to the dispersion, and W6(Φλ,ε(z)) =ε4BD/aL(z)+OK(ε2) uniformly in the resonance point z and the deformation parameter λ. Here L is the spectral diameter and B,D are the local cubic and mean-flow coefficients. Thus BD≠0 gives nonzero scattering on relatively open resonance sets. At the Craig--Guyenne--Sulem temporal coefficients, the corresponding exact Euler resonances satisfy μ/4W6E(1+μΨμ(z)) =−8/L(z)+OK(μ), where μ is the relative bandwidth. The leading term comes from nine contractions of quartic vertices; every other connected degree-six contribution is uniformly bounded. We classify the coefficient values for which the generic six-wave coefficient vanishes identically; in the temporal case, this also gives cancellation on every non-permutation four-wave resonance. Outside these sets, the Zakharov--Schulman condition restricts every regular C3 quadratic leading symbol to span{1,k,ω(k)}, the span of the symbols of wave action, momentum, and linear energy. Consequently the physical Dysthe models cannot support an inverse-scattering hierarchy with infinitely many linearly independent regular C3 quadratic leading symbols. The cancellation sets contain known nonlinear Schrödinger, mixed Chen--Lee--Liu, Calogero--Moser derivative nonlinear Schrödinger, and Hirota representatives.
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BibTeX
@misc{sutherland2026sixwave,
title = {{Six-wave scattering in Hamiltonian Dysthe equations: Euler matching and integrability obstructions}},
author = {Alex J. Sutherland and Solomon C. Yim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23175},
eprint = {2609.23175},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23175},
}
RIS
TY - UNPB
TI - Six-wave scattering in Hamiltonian Dysthe equations: Euler matching and integrability obstructions
AU - Alex J. Sutherland
AU - Solomon C. Yim
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.23175
UR - https://arxiv.org/abs/2609.23175
AB - We compute the effective six-wave interaction in spatial and temporal Hamiltonian Dysthe equations and compare it with the complete degree-six coefficient of the deep-water Euler Hamiltonian. On every compact subset K of a generic quadratic-dispersion resonance patch, exact resonances persist under a small cubic correction to the dispersion, and W_6(Φ_λ,ε(z)) =ε4BD/aL(z)+O_K(ε^2) uniformly in the resonance point z and the deformation parameter λ. Here L is the spectral diameter and B,D are the local cubic and mean-flow coefficients. Thus BD≠0 gives nonzero scattering on relatively open resonance sets. At the Craig--Guyenne--Sulem temporal coefficients, the corresponding exact Euler resonances satisfy μ/4W_6^E(1+μΨ_μ(z)) =−8/L(z)+O_K(μ), where μ is the relative bandwidth. The leading term comes from nine contractions of quartic vertices; every other connected degree-six contribution is uniformly bounded. We classify the coefficient values for which the generic six-wave coefficient vanishes identically; in the temporal case, this also gives cancellation on every non-permutation four-wave resonance. Outside these sets, the Zakharov--Schulman condition restricts every regular C^3 quadratic leading symbol to span{1,k,ω(k)}, the span of the symbols of wave action, momentum, and linear energy. Consequently the physical Dysthe models cannot support an inverse-scattering hierarchy with infinitely many linearly independent regular C^3 quadratic leading symbols. The cancellation sets contain known nonlinear Schrödinger, mixed Chen--Lee--Liu, Calogero--Moser derivative nonlinear Schrödinger, and Hirota representatives.
ER -