  The long-time asymptotics is analyzed for finite energy solutions of the 1D
Schr\"odinger equation coupled to a nonlinear oscillator. The coupled system is
invariant with respect to the phase rotation group U(1). For initial states
close to a solitary wave, the solution converges to a sum of another solitary
wave and dispersive wave which is a solution to the free Schr\"odinger
equation. The proofs use the strategy of Buslaev-Perelman: the linerization of
the dynamics on the solitary manifold, the symplectic orthogonal projection and
method of majorants.
