On Asymptotic Stability of Solitary Waves in Discrete Schrödinger Equation Coupled to Nonlinear Oscillator

Details On Asymptotic Stability of Solitary Waves in Discrete Schrödinger Equation Coupled to Nonlinear Oscillator On Asymptotic Stability of Solitary Waves in Discrete Schrödinger Equation Coupled to Nonlinear Oscillator

2008-05-22

arxiv.org/abs/0805.3403v1

Mathematics

Analysis of PDEs

22 pages

Scientific paper

10.1134/S1061920808040067

The long-time asymptotics is analyzed for finite energy solutions of the 1D discrete Schr\"odinger equation coupled to a nonlinear oscillator. The coupled system is invariant with respect to the phase rotation group. 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, method of majorants, etc.

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