On Asymptotic Stability of Solitary Waves in Discrete Schr\"odinger Equation Coupled to Nonlinear Oscillator

TL;DR

This paper analyzes the long-time behavior of solutions in a coupled discrete Schrödinger and nonlinear oscillator system, showing convergence to a solitary wave plus dispersive wave for initial states near a solitary wave.

Contribution

It extends the understanding of asymptotic stability for coupled discrete Schrödinger systems using advanced analytical techniques.

Findings

Solutions near a solitary wave converge to a solitary wave plus dispersive wave.

The system exhibits asymptotic stability under certain initial conditions.

The analysis employs the Buslaev-Perelman strategy for stability proofs.

Abstract

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.