Carles, Rémi Geometric optics and long range scattering for one-dimensional nonlinear Schrödinger equations. (English) Zbl 1029.35211 Commun. Math. Phys. 220, No. 1, 41-67 (2001). In this paper the scattering theory to the nonlinear Schrödinger equation \[ i \partial_t \psi + \frac{1}{2} \partial_x^2 \psi = \lambda |\psi|^2 \psi \] in one space dimension is considered for \(\lambda \in {\mathbb R}\). For this equation one cannot compare the nonlinear dynamics with the free dynamics. For, it is known that if \(\psi\) is a solution of this nonlinear equation and if \(U_0(-t)\psi (t)-\psi_- \rightarrow 0\) in \(L^2\) for \(t \rightarrow -\infty\), then \(\psi=\psi_-=0\). T. Ozawa [Commun. Math. Phys. 139, 479-493 (1991; Zbl 0970.35145)] defined a new operator such that the evolution of \(\psi_-\) under this dynamics can be compared to the asymptotic behavior of a certain solution of the nonlinear Schrödinger equation. In the article under review the author uses methods of geometrical optics introduced by him [Indiana Univ. Math. J. 49, 475-551 (2000; Zbl 0970.35145)] to rediscover this operator and to improve the convergence estimates. The new evolution operator is of the form \[ e^{iS^-(t)}U_0(t) \psi_- \] with the phase shift \[ S^-(t,x)=\frac{\lambda}{2\pi}\left |\hat{\psi}_-\left( \frac{x}{t} \right) \right |^2 \log t. \] Reviewer: Hans-Dieter Alber (Darmstadt) Cited in 16 Documents MSC: 35Q60 PDEs in connection with optics and electromagnetic theory 35P25 Scattering theory for PDEs 35Q55 NLS equations (nonlinear Schrödinger equations) 81U05 \(2\)-body potential quantum scattering theory 78A05 Geometric optics Keywords:scattering theory; nonlinear Schrödinger equation; nonlinear geometrical optics Citations:Zbl 0970.35145 PDFBibTeX XMLCite \textit{R. Carles}, Commun. Math. Phys. 220, No. 1, 41--67 (2001; Zbl 1029.35211) Full Text: DOI