Narrow Results

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 $Q_{\pm }$ for $x\to \pm \infty$. For special choices of $Q_{\pm }$, 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.