Asymptotic dynamics of nonlinear Schrödinger equations: Resonance-dominated and dispersion-dominated solutions
Article first published online: 7 NOV 2001
Copyright © 2002 John Wiley & Sons, Inc.
Communications on Pure and Applied Mathematics
Volume 55, Issue 2, pages 153–216, February 2002
How to Cite
Tsai, T.-P. and Yau, H.-T. (2002), Asymptotic dynamics of nonlinear Schrödinger equations: Resonance-dominated and dispersion-dominated solutions. Comm. Pure Appl. Math., 55: 153–216. doi: 10.1002/cpa.3012
- Issue published online: 3 DEC 2001
- Article first published online: 7 NOV 2001
- Manuscript Revised: JUL 2001
- Manuscript Received: NOV 2000
- NSF Grant. Grant Number: DMS-0072098
- Center for Theoretical, Taiwan
We consider a linear Schrödinger equation with a nonlinear perturbation in ℝ3. Assume that the linear Hamiltonian has exactly two bound states and its eigen-values satisfy some resonance condition. We prove that if the initial data is sufficiently small and is near a nonlinear ground state, then the solution approaches to certain nonlinear ground state as the time tends to infinity. Furthermore, the difference between the wave function solving the nonlinear Schrödinger equation and its asymptotic profile can have two different types of decay: The resonance-dominated solutions decay as t−1/2 or the dispersion-dominated solutions decay at least like t−3/2. © 2002 John Wiley & Sons, Inc.