**Communications on Pure and Applied Mathematics**

# Asymptotic dynamics of nonlinear Schrödinger equations: Resonance-dominated and dispersion-dominated solutions

## Abstract

We consider a linear Schrödinger equation with a nonlinear perturbation in ℝ^{3}. 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.