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Inverse scattering for the non‐linear Schrödinger equation: Reconstruction of the potential and the non‐linearity
Author(s) -
Weder Ricardo
Publication year - 2001
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.216
Subject(s) - mathematics , inverse scattering problem , scattering , mathematical physics , limit (mathematics) , linearity , inverse , inverse problem , operator (biology) , mathematical analysis , schrödinger equation , combinatorics , quantum mechanics , physics , geometry , biochemistry , chemistry , repressor , transcription factor , gene
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.