The generalized Zakharov–Shabat system and the soliton perturbations

The nonlinear evolution equations and their inhomogeneous versions related through the inverse scattering method to the generalized Zakharov–Shabat system $L=i d/dx + q(x) -\lambda J$ are studied. Here we assume that the potential $q(x)=[J,Q(x)]$ takes values in the simple Lie algebra $\mathfrak {g}$ and that $J$ is a nonregular element of the Cartan subalgebra $\mathfrak {h}$. The corresponding systems of equations for the scattering data of $L$ are derived. These can be applied to the study of soliton perturbations of such equations as the matrix nonlinear Schrödinger equation, the matrix $n$–wave equations etc.