$L^2$-Sobolev space bijectivity of the inverse scattering of a $3 \times 3$ AKNS system

TL;DR

This paper proves the bijectivity of the direct and inverse scattering transforms for a 3x3 AKNS system within certain Sobolev spaces, advancing the mathematical understanding of integrable systems related to the Manakov and Sasa-Satsuma equations.

Contribution

It establishes the $L^2$-Sobolev space bijectivity for the scattering map of the 3x3 AKNS system, extending previous results to higher-order Sobolev spaces.

Findings

Bijectivity of scattering transforms on $H^{i,1}( eal)$ for $i=1,2$

Extension of inverse scattering theory to 3x3 AKNS systems

Mathematical foundation for analyzing related integrable PDEs

Abstract

We study the $L^2$-Sobolev space bijectivity of the direct and inverse scattering of the $3\times 3$ AKNS system associated to the Manakov system and Sasa-Satsuma equation. We establish the bijectivity on the weighted Sobolev space $H^{i,1}(\mathbb{R})$ for $i=1,2$.