N-wave interactions related to simple Lie algebras. ℤ₂-reductions and soliton solutions

The reductions of the integrable N-wave type equations solvable by the inverse scattering method with the generalized Zakharov-Shabat systems L and related to some simple Lie algebra g are analysed. The Zakharov-Shabat dressing method is extended to the case when g is an orthogonal algebra. Several types of one-soliton solutions of the corresponding N-wave equations and their reductions are studied. We show that one can relate a (semi-)simple subalgebra of g to each soliton solution. We illustrate our results by four-wave equations related to so(5) which find applications in Stokes-anti-Stokes wave generation.