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On asymptotic stability of solitary waves in Schrödinger equation coupled to nonlinear oscillator. (English) Zbl 1185.35247

The long-time asymptotics is analysed for finite energy solutions of the 1D Schrödinger equation coupled to a nonlinear oscillator. The techniques of Buslaev and Perelman based on the symplectic geometry in Hilbert space and the spectral theory of nonselfadjoint operators are used. 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 Schrödinger equation.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q51 Soliton equations
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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