On Inverse scattering for the nonlocal nonlinear Schrodinger equation with slowly decaying data

Abstract

This paper discusses about an inverse scattering problem to the nonlocal nonlinear Schrodinger equation. We extend the previous result treated by Y. Zhao and E. Fan of inverse scattering result at the point with slowly decaying data. More precisely, we show the global existence of solution in $H^{1,s}$ for $s>1/2$ by assuming small data in $L^1$. The proof relies on the inverse scattering method based on the ZS-AKNS (Zakharov-Shabat, Ablowitz-Kaup-Newell-Segur) eigenvalue problem on the spectral theory of ordinary differential equations upon the framework of slowly decaying data.