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.

本文致力于$氮$-与辛李代数相关的具有恒定边界条件的波动方程。我们研究一类 Lax 算子的谱特性$L$，其潜力$问（X,t）$趋向于常数${问}_{\pm }$为了$X\to \pm \mathrm{无穷大}$。对于特殊选择${问}_{\pm }$，我们概述了光谱特性$L$，直接散射变换并构造其基本解析解。我们将 Wronskian 关系推广到 CBC 的情况——这使我们能够分析散射数据和$X$-势能的导数${问}_X$。接下来，使用朗斯基关系，我们推导出色散定律$氮$-wave 层次结构并描述与给定 Lax 算子相关的 NLEE。