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.

We analyze the algebraic aspects of solving the multicomponent nonlinear Schrödinger (MNLS) equations related to the symmetric spaces of BD.I-type. This analysis includes: i) the spectral properties of the MNLS equations under the nonvanishing (constant) boundary conditions; ii) the construction of new equations of MNLS type imposing additional reductions; iii) the involutivity of their integrals of motion proven by using the method of the classical R-matrix; iv) brief but explicit description of their hierarchies of Hamiltonian structures.