We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
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), Asterisque208 (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.
