On Plancherel's identity for a two-dimensional scattering transform
Abstract
We consider the \overline{\partial} -Dirac system that Ablowitz and Fokas used to transform the defocussing Davey-Stewartson system to a linear evolution equation. The nonlinear Plancherel identity for the associated scattering transform was established by Beals and Coifman for Schwartz functions. Sung extended the validity of the identity to functions belonging to L^1\cap L^∞({R}^2) and Brown to L2-functions with sufficiently small norm. More recently, Perry extended to the weighted Sobolev space H1,1({R}^2) and here we extend to Hs,s({R}^2) with 0 < s < 1.
- Publication:
-
Nonlinearity
- Pub Date:
- August 2015
- DOI:
- 10.1088/0951-7715/28/8/2721
- arXiv:
- arXiv:1503.00093
- Bibcode:
- 2015Nonli..28.2721A
- Keywords:
-
- Mathematics - Classical Analysis and ODEs;
- Mathematics - Analysis of PDEs;
- 35P25
- E-Print:
- 11 pages