## On the asymptotic stability of solitary waves in the discrete Schrödinger equation coupled to a nonlinear oscillator.(English)Zbl 1167.35515

Summary: 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, in: Nonlinear evolution equations. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 164 (22), 75–98 (1995; Zbl 0841.35108)]: the linearization of the dynamics on the solitary manifold, the symplectic orthogonal projection, method of majorants, etc.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems 39A12 Discrete version of topics in analysis

Zbl 0841.35108
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### References:

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