Physics – Mathematical Physics
Scientific paper
2002-05-10
Adv. Theor. Math. Phys. 6 (2002) 107-139
Physics
Mathematical Physics
to appear in Adv. Theor. Math. Phys
Scientific paper
We consider a nonlinear Schr\"odinger 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 Schr\"odinger equations.
Tsai Tai-Peng
Yau Horng-Tzer
No associations
LandOfFree
Classification of Asymptotic Profiles for Nonlinear Schrödinger Equations with Small Initial Data does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Classification of Asymptotic Profiles for Nonlinear Schrödinger Equations with Small Initial Data, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Classification of Asymptotic Profiles for Nonlinear Schrödinger Equations with Small Initial Data will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-382356