On the Reductions and Hamiltonian Structures of N-Wave Type Equations

Abstract

The reductions of the integrable $N$-wave type equations solvable by the inverse scattering method with the generalized Zakharov–Shabat system $L$ and related to some simple Lie algebra $\mathfrak{g}$ are analyzed. Special attention is paid to the $\mathbb{Z}_2$ and $\mathbb{Z}_2 \times \mathbb{Z}_2$-reductions including ones that can be embedded also in the Weyl group of $\mathfrak{g}$. The consequences of these restrictions on the properties of their Hamiltonian structures are analyzed on specific examples which find applications to nonlinear optics.