Relaxation of Excited States in Nonlinear Schr\"odinger Equations

Abstract

We consider a nonlinear Schr\"odinger equation in $\R^3$ with a bounded local potential. The linear Hamiltonian is assumed to have two bound states with the eigenvalues satisfying some resonance condition. Suppose that the initial data is small and is near some nonlinear {\it excited} state. We give a sufficient condition on the initial data so that the solution to the nonlinear Schr\"odinger equation approaches to certain nonlinear {\it ground} state as the time tends to infinity.