Tai-Peng Tsai and Horng-Tzer Yau
Asymptotic Dynamics of Nonlinear Schr\"odinger Equations: Resonance Dominating and Radiation Dominating Solutions
(311K, dvi)

ABSTRACT.  We consider a linear Schr\"odinger equation with a small nonlinear 
perturbation in $R^3$. Assume that the linear 
Hamiltonian has exactly two bound states and its eigenvalues satisfy 
some resonance condition. We prove that if the initial data 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 Schr\"odinger 
equation and its asymptotic profile can have two different types of 
decay: 1. The resonance dominating solutions decay as 
$t^{-1/2}$. 2. The radiation dominating solutions decay at least like 
$t^{-3/2}$.