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 d m / d x = z 2 [ J , m ] + ( z Q + P ) m dm/dx = {z^2}[J,m] + (zQ + P)m ; J J is diagonal and skew-Hermitian [ 8 , 10 , 12 ] [8,10,12] . Following the idea of R. Beals and R. R. Coifman, we estimate the wedge products of the columns of m m by L 2 {L^2} -norm of the potential ( Q , P ) [ 4 ] (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 ) (Q(x,0),\,P(x,0)) = ({Q_0},{P_0}) is of Schwartz class.