The generalized Zakharov-Shabat system and the soliton perturbations

Abstract

The nonlinear evolution equations and their inhomogeneous versions related through the inverse scattering method to the generalized Zakharov-Shabat systemL=id/dx+q(x)−λJ are studied. Here we assume that the potentialq(x)=[J,Q(x)] takes values in the simple Lie algebra g and thatJ is a nonregular element of the Cartan subalgebra ν. The corresponding systems of equations for the scattering data ofL are derived. These can be applied to the study of soliton perturbations of such equations as the matrix nonlinear Schrodinger equation, the matrixn-wave equations, etc.