Inverse scattering for the non‐linear Schrödinger equation: Reconstruction of the potential and the non‐linearity

In this paper we consider the inverse scattering problem for the non‐linear Schrödinger equation on the line \def\dr{{\rm d}}$$i{\partial\over\partial t}u(t,x)=‐{\dr^2\over\dr x^2}u(t,x)+V_0(x)u(t,x)+\sum_{j=1}^{\infty}V_j(x)|u|^{2(j_0+j)}u(t,x)$$\nopagenumbers\end We prove, under appropriate conditions, that the small‐amplitude limit of the scattering operator determines uniquely V j , j =0,1,… . Our proof gives also a method for the reconstruction of the V j , j =0,1,… . Copyright © 2001 John Wiley & Sons, Ltd.