Asymptotic Dynamics of Nonlinear Schrödinger Equations: Resonance Dominated and Radiation Dominated Solutions

We consider a linear Schrödinger equation with a small nonlinear perturbation in R3. 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 Schrödinger equation and its asymptotic profile can have two different types of decay: 1. The resonance dominated solutions decay as t−1/2. 2. The radiation dominated solutions decay at least like t−3/2. [email protected] Work partially supported by NSF grant DMS-0072098, [email protected]