N-wave interactions related to simple Lie algebras. Z_2- 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 analyzed. 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 to each soliton solution one can relate a (semi-)simple subalgebra of g. We illustrate our results by 4-wave equations related to so(5) which find applications in Stockes-anti-Stockes wave generation.Comment: 18 pages, 1 figure, LaTeX 2e, IOP-style; More clear exposition. Introduction and Section 5 revised. Some typos are correcte

On Kaup-Kupershchmidt–type equations and their soliton solutions

We start with the Lax representation for the Kaup-Kupersschmidt equation (KKE). We outline the deep relation between the scalar Lax operator and the matrix Lax operators related to Kac-Moody algebras. Then we derive the MKdV equations gauge equivalent to the KKE. Next we outline the symmetry and the spectral properties of the relevant Lax operator. Using the dressing Zakharov-Shabat method we demonstrate that the MKdV and KKE have two types of onesoliton solutions and briefly comment on their properties

On the Caudrey-Beals-Coifman System and the Gauge Group Action

The generalized Zakharov-Shabat systems with complex-valued Cartan elements and the systems studied by Caudrey, Beals and Coifman (CBC systems) and their gauge equivalent are studies. This includes: the properties of fundamental analytical solutions (FAS) for the gauge-equivalent to CBC systems and the minimal set of scattering data; the description of the class of nonlinear evolutionary equations solvable by the inverse scattering method and the recursion operator, related to such systems; the hierarchies of Hamiltonian structures.Comment: 12 pages, no figures, contribution to the NEEDS 2007 proceedings (Submitted to J. Nonlin. Math. Phys.