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

Abstract

We consider the scattering transform for the first order system in the plane,We show that the scattering map is Lipschitz continuous on a neighborhood of zero in L 2 .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 byand ∂x and ∂ x are the standard derivatives with respect to x = x 1 + ix 2 andx. The entries of the matrix Q, q 1 (x) and q 2 (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).

Inverse Scattering and Global Well-Posedness in One and Two Space Dimensions

“…(ii) |pSpqqq pkq| ď C p}q} L 2 q |Mp qpkq| Theorem 1.9 considerably extends earlier work of Brown [11] and Perry [38], who considered the scattering map respectively for small data in L 2 pR 2 q and data in a weighted space H 1,1 pR 2 q analogous to the space H 1,1 pRq for the NLS. It also illuminates other work of Astala-Faraco-Rogers [4] and Brown-Ott-Perry [12] on the Fourier-like mapping properties of S. The maximal function estimate is particularly important for the analysis of scattering since it implies that the solution of DSII by inverse scattering is bounded pointwise by a maximal function for the solution of the linear problem.…”

Estimates for a family of multi-linear forms

“…As part of this recovery, it is interesting to know something about the continuity properties of the scattering map. This was one motivation for the work of Brown [8]. This work of Brown shows that the scattering map is continuous in a neighborhood 0 in L 2 .…”

On Plancherel's identity for a two-dimensional scattering transform

“…Unlike in the linear case, the Lipschitz continuity is not an immediate consequence of the identity and so this is not enough to extend to L 2 . Sung treated F ∈ L 1 ∩ L ∞ (R 2 ) in [18], then Brown [9] proved Lipschitz continuity for L 2 -functions with sufficiently small norm, Mathematics Subject Classification. Primary 35P25, 45Q05; Secondary 42B37.…”