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Inverse scattering for the nonlinear Schrödinger equation II. Reconstruction of the potential and the nonlinearity in the multidimensional case


Author: Ricardo Weder
Journal: Proc. Amer. Math. Soc. 129 (2001), 3637-3645
MSC (2000): Primary 35R30, 35Q55, 35P25, 81U40
Published electronically: April 25, 2001
MathSciNet review: 1860498
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Abstract:

We solve the inverse scattering problem for the nonlinear Schrödinger equation on ${\mathbf R}^n, n \geq 3$: \begin{equation*}i \frac{\partial }{\partial t}u(t,x)= -\Delta u(t,x)+V_0(x)u(t,x) + \sum_{j=1}^{\infty} V_j(x)\vert u\vert^{2(j_0+j)} u(t,x). \end{equation*}

We prove that the small-amplitude limit of the scattering operator uniquely determines $V_{j}, j=0,1, \cdots $. Our proof gives a method for the reconstruction of the potentials $V_{j}, j=0,1, \cdots $. The results of this paper extend our previous results for the problem on the line.


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Additional Information

Ricardo Weder
Affiliation: Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Apartado Postal 20-726, México D.F. 01000
Email: weder@servidor.unam.mx

DOI: https://doi.org/10.1090/S0002-9939-01-06016-6
Received by editor(s): January 19, 2000
Received by editor(s) in revised form: April 27, 2000
Published electronically: April 25, 2001
Additional Notes: This research was partially supported by Proyecto PAPIIT-DGAPA IN 105799.
The author is a Fellow of Sistema Nacional de Investigadores.
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2001 by the author