$N$-Wave Type Systems and their Gauge Equivalent Related to the Orthogonal Algebras

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 $\mathbb{Z}_2$-reductions including ones that can be embedded also in the Weyl group of $\mathfrak{g}$. The consequences of these restrictions on the structure of the dresing factors are outlined. An example of 4-wave equations (with application to nonlinear optics) and its gauge equivalent are given.