Inverse scattering for the nonlinear Schrödinger equation II. Reconstruction of the potential and the nonlinearity in the multidimensional case
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- by Ricardo Weder PDF
- Proc. Amer. Math. Soc. 129 (2001), 3637-3645
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)|u|^{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.References
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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
- 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
- © Copyright 2001 by the author
- Journal: Proc. Amer. Math. Soc. 129 (2001), 3637-3645
- MSC (2000): Primary 35R30, 35Q55, 35P25, 81U40
- DOI: https://doi.org/10.1090/S0002-9939-01-06016-6
- MathSciNet review: 1860498