Six-wave scattering in Hamiltonian Dysthe equations: Euler matching and integrability obstructions

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 $$W_6({\mathrm{\Phi }}_{\lambda ,\epsilon }(z))=\epsilon \frac{4BD}{aL(z)}+O_K({\epsilon }^2)$$ uniformly in the resonance point $z$ and the deformation parameter $\lambda$. Here $L$ is the spectral diameter and $B,D$ are the local cubic and mean-flow coefficients. Thus $BD\mathrm{\ne }0$ gives nonzero scattering on relatively open resonance sets. At the Craig–Guyenne–Sulem temporal coefficients, the corresponding exact Euler resonances satisfy $$\frac{\mu }{4}{W}_6^{\mathrm{E}}(1+\mu {\mathrm{\Psi }}_{\mu }(z))=-\frac{8}{L(z)}+O_K(\mu ),$$ where $\mu$ 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 $\mathrm{span}\{1,k,\omega (k)\}$, 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.