On asymptotic stability of solitary waves in discrete Klein-Gordon equation coupled to a nonlinear oscillator

The long-time asymptotics is analysed for finite energy solutions of the 1D discrete Klein-Gordon 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 Klein-Gordon equation. The proofs develop the strategy of Buslaev-Perelman: the linearization of the dynamics on the solitary manifold, the symplectic orthogonal projection, method of majorants, etc.