Estimates for the Scattering Map Associated with a Two-Dimensional First-Order System

We consider the scattering transform for the first order system in the plane, ( ∂x̄ 0 0 ∂x ) ψ − ( 0 q1 q2 0 ) ψ = 0. We show that the scattering map is Lipschitz continuous on a neighborhood of zero in L2. This paper gives an estimate for the scattering map associated to a first-order system Dψ −Qψ = 0 (1) in the plane. Here, D and Q are defined by D = ( ∂x̄ 0 0 ∂x ) and Q = ( 0 q q 0 ) and ∂x̄ and ∂x are the standard derivatives with respect to x = x 1 + ix and x̄. The entries of the matrix Q, q(x) and q(x) are complex valued functions on the complex plane. (We will consistently use superscripts to indicate components.) The system (1) was studied by Beals and Coifman [2, 3], and a number of other authors (see Fokas and Ablowitz [7] for an earlier formal treatment, Sung [13, 14, 15] for a detailed rigorous treatment and the review articles [4, 6] for additional references). ∗Author supported in part by the National Science Foundation, Division of Mathematical Sciences.