Scattering and localized states for defocusing nonlinear Schr\'odinger equations with potential
- Resource Type
- Working Paper
- Authors
- Soffer, Avy; Stewart, Gavin
- Source
- Subject
- Mathematics - Analysis of PDEs
Mathematical Physics
35Q55, 35Q41 (Primary) 35B40 (Secondary)
- Language
We study the large-time behavior of global energy class solutions of the one dimensional nonlinear Schr\"odinger equation with a general localized potential term and a defocusing nonlinear term. By using a new type of interaction Morawetz estimate localized to an exterior region, we prove that these solutions decompose into a free wave and a weakly localized part which is asymptotically orthogonal to any fixed free wave. We further show that the $L^2$ norm of this weakly localized part is concentrated in the region $|x| \leq t^{1/2+}$, and that the energy ($\dot{H}^1$) norm is concentrated in $|x| \leq t^{1/3+}$.
Comment: Typos fixed. 32 pages