Five-wave classical scattering matrix and integrable equations

We study the five-wave classical scattering matrix for nonlinear and dispersive Hamiltonian equations with a nonlinearity of the type $u\, \partial u/\partial x$. Our aim is to find the most general nontrivial form of the dispersion relation $\omega(k)$ for which the five-wave interaction scattering matrix is identically zero on the resonance manifold. As could be expected, the matrix in one dimension is zero for the Korteweg–de Vries equation, the Benjamin–Ono equation, and the intermediate long-wave equation. In two dimensions, we find a new equation that satisfies our requirement.