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

Abstract

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.

Keywords:

• long-time asymptotics
• discrete Klein–Gordon equation
• nonlinear oscillator
• solitary wave

AMS Subject Classifications::

• 39A14
• 39A30

Acknowledgement

The author was partly supported by the grants of FWF P19138-N13, DFG 436 RUS 113/929/0-1, and RFBR.