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

Weder, Ricardo

doi:10.1090/S0002-9939-01-06016-6

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

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