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

Abstract

This paper is devoted to [Formula: see text]-wave equations with constant boundary conditions related to symplectic Lie algebras. We study the spectral properties of a class of Lax operators [Formula: see text], whose potentials [Formula: see text] tend to constants [Formula: see text] for [Formula: see text]. For special choices of [Formula: see text], we outline the spectral properties of [Formula: see text], 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 [Formula: see text]-derivative of the potential [Formula: see text]. Next, using the Wronskian relations, we derive the dispersion laws for the [Formula: see text]-wave hierarchy and describe the NLEE related to the given Lax operator.