## 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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### References:

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