On asymptotic stability of solitons in a nonlinear Schrödinger equation

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

On asymptotic stability of solitons in a nonlinear Schrödinger equation

Alexander Komech ^1, , Elena Kopylova ^2, and David Stuart ^3,

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

Abstract
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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.

Keywords: Schrödinger equation, Asymptotic stability, solitary waves..

Mathematics Subject Classification: 35Q55, 37K4.

Citation: Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure and 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., \textbf{33} (2008), 669-705. doi: 10.1080/03605300801970937.
[2]	V. Buslaev and C. Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 20 (2003), 419-475. 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), 164-173. 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), 225-252. 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, arXiv:0807.1878.
[6]	M. Merkli and I. M. Sigal, A time-dependent theory of quantum resonances, Commun. Math. Phys., 201 (1999), 549-576. doi: 10.1007/s002200050568.
[7]	R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Commun. Math. Phys., 164 (1994), 305-349. 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), 310-326. 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), 9-74. 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), 977-1071. doi: 10.1142/S0129055X04002175.

show all references

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., \textbf{33} (2008), 669-705. doi: 10.1080/03605300801970937.
[2]	V. Buslaev and C. Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 20 (2003), 419-475. 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), 164-173. 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), 225-252. 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, arXiv:0807.1878.
[6]	M. Merkli and I. M. Sigal, A time-dependent theory of quantum resonances, Commun. Math. Phys., 201 (1999), 549-576. doi: 10.1007/s002200050568.
[7]	R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Commun. Math. Phys., 164 (1994), 305-349. 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), 310-326. 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), 9-74. 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), 977-1071. doi: 10.1142/S0129055X04002175.

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