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MiS Preprint
37/2007

On Asymptotic Stability of Solitary Waves in a Nonlinear Schrödinger Equation

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

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.

Received:
Apr 10, 2007
Published:
Apr 10, 2007

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2008 Repository Open Access
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On asymptotic stability of solitary waves in Schrödinger equation coupled to nonlinear oscillator

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