On Asymptotic Stability of Solitary Waves in Schrödinger Equation Coupled to Nonlinear Oscillator
نویسندگان
چکیده
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 [2, 3]: the linerization of the dynamics on the solitary manifold, the symplectic orthogonal projection, method of majorants, etc. Supported partly by RFBR research grants 05-0101076 and 05-01002944. On leave Institute of the Information Transmission Problems RAS. Supported partly by Alexander von Humboldt Research Award, RFBR grant 07-01-00018a, and Max-Planck Institute for Mathematics in the Sciences (Leipzig). Supported partly by FWF grant P19138-N13, DFG grant 436 RUS 113/929/0-1, and RFBR grant 06-0100096. Partially supported by EPSRC grant A00133/01
منابع مشابه
On Asymptotic Stability of Solitary Waves in Discrete Schrödinger Equation Coupled to Nonlinear Oscillator
The long-time asymptotics is analyzed for finite energy solutions of the 1D discrete 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. T...
متن کاملGlobal well-posedness for the Schrödinger equation coupled to a nonlinear oscillator
The Schrödinger equation with the nonlinearity concentrated at a single point proves to be an interesting and important model for the analysis of long-time behavior of solutions, such as the asymptotic stability of solitary waves and properties of weak global attractors. In this note, we prove global well-posedness of this system in the energy space H1.
متن کاملOn Asymptotic Stability of Solitary Waves in a Nonlinear Schrödinger Equation
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...
متن کاملSimplest Equation Method for nonlinear solitary waves in Thomas- Fermi plasmas
The Thomas-Fermi (TF) equation has proved to beuseful for the treatment of many physical phenomena. In this pa-per, the traveling wave solutions of the KdV equation is investi-gated by the simplest equation method. Also, the effect of differentparameters on these solitary waves is considered. The numericalresults is conformed the good accuracy of presented method.
متن کاملAsymptotic Stability and Completeness in the Energy Space for Nonlinear Schrödinger Equations with Small Solitary Waves
In this paper, we study a class of nonlinear Schrödinger equations (NLS) which admit families of small solitary wave solutions. We consider solutions which are small in the energy space H, and decompose them into solitary wave and dispersive wave components. The goal is to establish the asymptotic stability of the solitary wave and the asymptotic completeness of the dispersive wave. That is, we...
متن کامل