×

\(N\)-wave interactions related to simple Lie algebras. \(\mathbb{Z}_2\)-reductions and soliton solutions. (English) Zbl 0988.35143

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.

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
PDFBibTeX XMLCite
Full Text: DOI arXiv