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Scattering for the \(L^2\) supercritical point NLS. (English) Zbl 1457.35064

In the paper [M. I. Molina and C. A. Bustamante, “The attractive nonlinear delta-function potential”, Am. J. Phys. 70, No. 1, 67 (2002; doi:10.1119/1.1417529], it is introduced a model for wave propagation in a 1D linear medium with a narrow strip of non-linear material. This model corresponds to the NLS equation \[\begin{cases} i\partial_t\psi+\partial_x^2\psi+K(x)|\psi|^{p-1}\psi=0&t,x\in\mathbb R\\ \psi(x,0)=\psi_0(x)\end{cases}\tag{\(\ast\)}\] Here \(K(x)=\delta_0\) is the Dirac delta function supported at \(0\) and the exponent \(p\) is assumed \(>1\).
The main result of the paper under review is the following. Assuming conditions on the initial data, \(\psi_0(x)\), related to the function \(\varphi_0(x):=2e^{\frac{1}{p-1}}e^{-|x|}\) (this is the ground state stationary solution of \((*)\)), there exist functions \(\psi^+,\psi^-\in H^1(\mathbb R)\) such that, if \(\psi(t,x)\) is a solution of \((*)\) and \(\psi(t,\cdot)\in H^1(\mathbb R)\), then \[\lim_{t\to\pm\infty}\| e^{-it\partial_x^2}\psi(t,\cdot)-\psi^\pm(\cdot)\|_{H^1(\mathbb R)}=0\,.\]

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
35P25 Scattering theory for PDEs
78A60 Lasers, masers, optical bistability, nonlinear optics
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