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

Abstract

Abstract.In this paper we derive some results for the Zakharov-Shabat system of the form dmfdx = z2[J, m] + (zQ + P)m ; J is diagonal and skewHermitian [8,10, 12]. Following the idea of R. Beals and R. R. Coifman, we estimate the wedge products of the columns of m by L2-norm of the potential (Ö. 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)) = (ßo. Po) is of Schwartz class. Direct and inverse scattering for the Z-S system WITH A QUADRATIC PARAMETERBy a potential here we mean a pair of functions (Q,P): R -► Mn(C) =set of « x « complex matrices; Q is off-diagonal and the diagonal part of P equals the diagonal part of Q(adJ)~xQ, and Q, Qx , P, Px E Lx . We