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
$t^{-3/2}$.