On the asymptotic stability of solitary waves in the discrete Schrödinger equation coupled to a nonlinear oscillator

The long-time asymptotics is analyzed for finite energy solutions of the 1D discrete Schrödinger 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 a dispersive wave which is a solution to the free Schrödinger equation. The proofs use the strategy of Buslaev-Perelman [V.S. Buslaev, G.S. Perelman, On the stability of solitary waves for nonlinear Schrödinger equations, Amer. Math. Soc. Trans. 2 164 (1995), 75-98]: the linearization of the dynamics on the solitary manifold, the symplectic orthogonal projection, method of majorants, etc.