Classification of Asymptotic Profiles for Nonlinear Schrödinger Equations with Small Initial Data
| dc.creator | Tsai, Tai-Peng | |
| dc.creator | Yau, Horng-Tzer | |
| dc.date | 2002-05-10 | |
| dc.date | 2003-12-26 | |
| dc.date.accessioned | 2026-07-07T04:29:11Z | |
| dc.date.available | 2026-07-07T04:29:11Z | |
| dc.description | We consider a nonlinear Schrödinger equation with a bounded local potential in $R^3$. The linear Hamiltonian is assumed to have two bound states with the eigenvalues satisfying some resonance condition. Suppose that the initial data are localized and small in $H^1$. We prove that exactly three local-in-space behaviors can occur as the time tends to infinity: 1. The solutions vanish; 2. The solutions converge to nonlinear ground states; 3. The solutions converge to nonlinear excited states. We also obtain upper bounds for the relaxation in all three cases. In addition, a matching lower bound for the relaxation to nonlinear ground states was given for a large set of initial data which is believed to be generic. Our proof is based on outgoing estimates of the dispersive waves which measure the relevant time-direction dependent information of the dispersive wave. These estimates, introduced in [16], provides the first general notion to measure the out-going tendency of waves in the setting of nonlinear Schrödinger equations. | |
| dc.description | to appear in Adv. Theor. Math. Phys | |
| dc.identifier | https://arxiv.org/abs/math-ph/0205015 | |
| dc.identifier | http://arxiv.org/abs/math-ph/0205015 | |
| dc.identifier | Adv. Theor. Math. Phys. 6 (2002) 107-139 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/57063 | |
| dc.subject | Mathematical Physics | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 35Q40, 35Q55 | |
| dc.title | Classification of Asymptotic Profiles for Nonlinear Schrödinger Equations with Small Initial Data | |
| dc.type | text |