On the Reductions and Scattering Data for the CBC System

Abstract

The reductions for the first order linear systems of the type: \[ L \psi (x,\lambda) \equiv \left(\mathbf{i} \frac{\rm d}{{\rm d}x} + q(x) - \lambda J \right) \psi (x,\lambda) = 0,\ J \in \mathfrak{h},\ q(x) \in \mathfrak{g}_J \] are studied. This system generalizes the Zakharov–Shabat system and the systems studied by Caudrey, Beals and Coifman (CBC systems). Here $J$ is a regular complex constant element of the Cartan subalgebra $\mathfrak{h} \subset \mathfrak{g}$ of the simple Lie algebra $\mathfrak{g}$ and the potential $q(x)$ takes values in the image $\mathfrak{g}_J$ of ad $_J$. Special attention is paid to the scattering data of CBC systems and their behaviour under the Weyl group reductions. The analytical properties of the generating functional of the integrals of motion and their reduced analogs are studied. These results are demonstrated on an example of $N$-wave type equations.