# American Institute of Mathematical Sciences

May  2012, 11(3): 1063-1079. doi: 10.3934/cpaa.2012.11.1063

## On asymptotic stability of solitons in a nonlinear Schrödinger equation

 1 Faculty of Mathematics of Vienna University, Vienna, Australia 2 IITP RAS, Moscow, Russian Federation 3 Centre for Mathematical Sciences, Cambridge, United Kingdom

Received  November 2010 Revised  May 2011 Published  December 2011

The long-time asymptotics is analyzed for finite energy solutions of the 1D Schrödinger equation coupled to a nonlinear oscillator through a localized nonlinearity. The coupled system is $U(1)$ invariant. This article, which extends the results of a previous one, provides a proof of asymptotic stability of the solitary wave solutions in the case that the linearization contains a single discrete oscillatory mode satisfying a non-degeneracy assumption of the type known as the Fermi Golden Rule.
Citation: Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063
##### References:
 [1] V. Buslaev, A. Komech, E. Kopylova and D. Stuart, On asymptotic stability of solitary waves in a nonlinear Schrödinger equation,, Comm. Partial Diff. Eqns., (2008), 669. doi: 10.1080/03605300801970937. [2] V. Buslaev and C. Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations,, Ann. Inst. Henri Poincaré, 20 (2003), 419. doi: 10.1016/S0294-1449(02)00018-5. [3] A. I. Komech and A. A. Komech, Global well-posedness for the Schrödinger equation coupled to a nonlinear oscillator,, Russ. J. Math. Phys., 14 (2007), 164. doi: 10.1134/S1061920807020057. [4] A. Komech and E.Kopylova, On Asymptotic stability of moving kink for relativistic Ginsburg-Landau equation,, Commun. Math. Phys., 302 (2011), 225. doi: 10.1007/s00220-010-1184-7. [5] A. Komech, E. Kopylova and D. Stuart, On asymptotic stability of solitons for nonlinear Schrödinger equation,, preprint, (). [6] M. Merkli and I. M. Sigal, A time-dependent theory of quantum resonances,, Commun. Math. Phys., 201 (1999), 549. doi: 10.1007/s002200050568. [7] R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves,, Commun. Math. Phys., 164 (1994), 305. doi: 10.1007/BF02101705. [8] C. A. Pillet and C. E. Wayne, Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations,, J. Diff. Eqns, 141 (1997), 310. doi: 10.1006/jdeq1997.3345. [9] A. Soffer and M. I. Weinstein, Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations,, Invent. Math., 136 (1999), 9. doi: 10.1007/s002220050303. [10] A. Soffer and M. I. Weinstein, Selection of the ground state for nonlinear Schrodinger equations,, Rev. Math. Phys., 16 (2004), 977. doi: 10.1142/S0129055X04002175.

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##### References:
 [1] V. Buslaev, A. Komech, E. Kopylova and D. Stuart, On asymptotic stability of solitary waves in a nonlinear Schrödinger equation,, Comm. Partial Diff. Eqns., (2008), 669. doi: 10.1080/03605300801970937. [2] V. Buslaev and C. Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations,, Ann. Inst. Henri Poincaré, 20 (2003), 419. doi: 10.1016/S0294-1449(02)00018-5. [3] A. I. Komech and A. A. Komech, Global well-posedness for the Schrödinger equation coupled to a nonlinear oscillator,, Russ. J. Math. Phys., 14 (2007), 164. doi: 10.1134/S1061920807020057. [4] A. Komech and E.Kopylova, On Asymptotic stability of moving kink for relativistic Ginsburg-Landau equation,, Commun. Math. Phys., 302 (2011), 225. doi: 10.1007/s00220-010-1184-7. [5] A. Komech, E. Kopylova and D. Stuart, On asymptotic stability of solitons for nonlinear Schrödinger equation,, preprint, (). [6] M. Merkli and I. M. Sigal, A time-dependent theory of quantum resonances,, Commun. Math. Phys., 201 (1999), 549. doi: 10.1007/s002200050568. [7] R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves,, Commun. Math. Phys., 164 (1994), 305. doi: 10.1007/BF02101705. [8] C. A. Pillet and C. E. Wayne, Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations,, J. Diff. Eqns, 141 (1997), 310. doi: 10.1006/jdeq1997.3345. [9] A. Soffer and M. I. Weinstein, Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations,, Invent. Math., 136 (1999), 9. doi: 10.1007/s002220050303. [10] A. Soffer and M. I. Weinstein, Selection of the ground state for nonlinear Schrodinger equations,, Rev. Math. Phys., 16 (2004), 977. doi: 10.1142/S0129055X04002175.
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