Gerdjikov, V. S.; Grahavoski, G. G.; Ivanov, R. I.; Kostov, N. A. \(N\)-wave interactions related to simple Lie algebras. \(\mathbb{Z}_2\)-reductions and soliton solutions. (English) Zbl 0988.35143 Inverse Probl. 17, No. 4, 999-1015 (2001). Summary: 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 \({\mathfrak g}\) are analysed. The Zakharov-Shabat dressing method is extended to the case when \({\mathfrak g}\) 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 \({\mathfrak g}\) 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. Cited in 1 ReviewCited in 17 Documents MSC: 35Q53 KdV equations (Korteweg-de Vries equations) 37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures Keywords:wave type equations; simple Lie algebra; Zakharov-Shabat dressing method; one-soliton solutions PDFBibTeX XMLCite \textit{V. S. Gerdjikov} et al., Inverse Probl. 17, No. 4, 999--1015 (2001; Zbl 0988.35143) Full Text: DOI arXiv