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Inverse scattering of the nonlinear Schrödinger equation with cubic convolution nonlinearity. (English) Zbl 1005.35093

Summary: It is shown that a potential \(V(x)\) and a constant \(\lambda\) are uniquely determined from the scattering operator \(S\) associated with the nonlinear Schrödinger equation \[ i{\partial u\over \partial t}+(-\Delta+ V)u+\lambda \bigl(|x|^{-\sigma}*|u|^2 \bigr)u=0, \] and the corresponding unperturbed equation \(i{\partial u\over \partial t}-\Delta u=0\).

MSC:

35R30 Inverse problems for PDEs
81U40 Inverse scattering problems in quantum theory
35P25 Scattering theory for PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
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