On the N-waves hierarchy with constant boundary conditions. Spectral properties

This paper is devoted to N-wave equations with constant boundary conditions related to symplectic Lie algebras. We study the spectral properties of a class of Lax operators L, whose potentials Q(x,t) tend to constants Q(+/-) for x ->+/-infinity. For special choices of Q(+/-), we outline the spectral properties of L, the direct scattering transform and construct its fundamental analytic solutions. We generalize Wronskian relations for the case of CBC - this allows us to analyze the mapping between the scattering data and the x-derivative of the potential Q(x). Next, using the Wronskian relations, we derive the dispersion laws for the N-wave hierarchy and describe the NLEE related to the given Lax operator.