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MiS Preprint

On Asymptotic Stability of Solitary Waves in a Nonlinear Schrödinger Equation

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


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.

Apr 10, 2007
Apr 10, 2007

Related publications

2008 Repository Open Access
V. S. Buslaev, Alexander Komech, E. A. Kopylova and D. Stuart

On asymptotic stability of solitary waves in Schrödinger equation coupled to nonlinear oscillator

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