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

摘要

We consider a linear Schr\"odinger equation with a small nonlinearperturbation in $R^3$. Assume that the linear Hamiltonian has exactly two boundstates and its eigenvalues satisfy some resonance condition. We prove that ifthe initial data is near a nonlinear ground state, then the solution approachesto certain nonlinear ground state as the time tends to infinity. Furthermore,the difference between the wave function solving the nonlinear Schr\"odingerequation and its asymptotic profile can have two different types of decay: 1.The resonance dominated solutions decay as $t^{-1/2}$. 2. The radiationdominated solutions decay at least like $t^{-3/2}$.