On the Dissipative Evolution Equations Associated with the Zakharov-Shabat System with a Quadratic Spectral Parameter

Abstract

In this paper we derive some results for the Zakharov-Shabat system of the form $dm/dx = {z^2}[J,m] + (zQ + P)m$; $J$ is diagonal and skew-Hermitian $[8,10,12]$. Following the idea of R. Beals and R. R. Coifman, we estimate the wedge products of the columns of $m$ by ${L^2}$-norm of the potential $(Q,P) [4]$. By this result we have the global existence of the dissipative evolution equations associated with this spectral problem if the generic initial data $(Q(x,0), P(x,0)) = ({Q_0},{P_0})$ is of Schwartz class.