Scattering and inverse scattering for the AKNS system: A rational function approach. Issue 4 (22nd August 2021)

• Record Type: Journal Article
• Authors: Trogdon, Thomas
• Is Part Of: Studies in applied mathematics. Volume 147:Issue 4(2021)
• Page Start: 1443
• Page End: 1480
• Publication Date: 2021-08-22
• Subjects: Cauchy integrals -- inverse scattering -- rational approximation; Mathematics -- Periodicals; Mathématiques -- Périodiques; Mathématique; Mathématique appliquée; Ressource Internet (Descripteur de forme); Périodique électronique (Descripteur de forme); 519
• DOI: 10.1111/sapm.12434
• Languages: English
• ISSNs: 0022-2526

Abstract

Abstract: We consider the use of rational basis functions to compute the scattering and inverse scattering transforms associated with the AKNS (Ablowitz–Kaup–Newell–Segur) system. The proposed numerical forward scattering transform computes the solution of the AKNS system that is valid on the entire real axis and thereby computes a reflection coefficient at a point by solving a single linear system. The proposed numerical inverse scattering transform makes use of a novel improvement in the rational function approach to the oscillatory Cauchy operator, enabling the efficient solution of certain Riemann–Hilbert problems without contour deformations. The latter development enables access to high‐precision computations and this is demonstrated on the inverse scattering transform for the one‐dimensional Schrödinger operator with a sech 2 potential.