On Asymptotic Stability of Solitary Waves in a Nonlinear Schrödinger Equation

V. S. Buslaev, Alexander Komech, E. A. Kopylova, and D. Stuart

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Submission date: 10. Apr. 2007
Pages: 38
published in: Communications in partial differential equations, 33 (2008) 4, p. 669-705 
DOI number (of the published article): 10.1080/03605300801970937
Bibtex
with the following different title: On asymptotic stability of solitary waves in Schrödinger equation coupled to nonlinear oscillator
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Abstract:
The long-time asymptotics is analyzed for finite energy solutions of the 1D 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 dispersive wave which is a solution to the free Schrödinger equation. The proofs use the strategy of Buslaev-Perelman [1,2]: the linerization of the dynamics on the solitary manifold, the symplectic orthogonal projection and method of majorants.

[1] V.S. Buslaev, G.S. Perelman, On nonlinear scattering of states which are close to a soliton, pp. 49-63 in: Méthodes Semi-Classiques, Vol.2 Colloque International (Nantes, juin 1991), Asterisque 208 (1992).
[2] V.S. Buslaev, G.S. Perelman, Scattering for the nonlinear Schrödinger equation: states close to a soliton, St. Petersburg Math. J. 4 (1993), 1111-1142.