On asymptotic stability of solitons in a nonlinear Schrödinger equation

Previous Article
Low regularity well-posedness for the 3D Klein - Gordon - Schrödinger system
CPAA Home
This Issue
Next Article
A faithful symbolic extension

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
Related Papers

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, ().
[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, ().
[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.

[1]	Sevdzhan Hakkaev. Orbital stability of solitary waves of the Schrödinger-Boussinesq equation. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1043-1050. doi: 10.3934/cpaa.2007.6.1043
[2]	Santosh Bhattarai. Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1789-1811. doi: 10.3934/dcds.2016.36.1789
[3]	Khaled El Dika. Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 583-622. doi: 10.3934/dcds.2005.13.583
[4]	Nghiem V. Nguyen, Zhi-Qiang Wang. Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schrödinger system. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1005-1021. doi: 10.3934/dcds.2016.36.1005
[5]	H. Kalisch. Stability of solitary waves for a nonlinearly dispersive equation. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 709-717. doi: 10.3934/dcds.2004.10.709
[6]	Reika Fukuizumi. Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 525-544. doi: 10.3934/dcds.2001.7.525
[7]	François Genoud. Existence and stability of high frequency standing waves for a nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1229-1247. doi: 10.3934/dcds.2009.25.1229
[8]	Juan Belmonte-Beitia, Vladyslav Prytula. Existence of solitary waves in nonlinear equations of Schrödinger type. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 1007-1017. doi: 10.3934/dcdss.2011.4.1007
[9]	David Usero. Dark solitary waves in nonlocal nonlinear Schrödinger systems. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 1327-1340. doi: 10.3934/dcdss.2011.4.1327
[10]	Daniele Cassani, João Marcos do Ó, Abbas Moameni. Existence and concentration of solitary waves for a class of quasilinear Schrödinger equations. Communications on Pure and Applied Analysis, 2010, 9 (2) : 281-306. doi: 10.3934/cpaa.2010.9.281
[11]	Tetsu Mizumachi, Dmitry Pelinovsky. On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (5) : 971-987. doi: 10.3934/dcdss.2012.5.971
[12]	Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215
[13]	Steve Levandosky, Yue Liu. Stability and weak rotation limit of solitary waves of the Ostrovsky equation. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 793-806. doi: 10.3934/dcdsb.2007.7.793
[14]	Alex H. Ardila. Stability of standing waves for a nonlinear SchrÖdinger equation under an external magnetic field. Communications on Pure and Applied Analysis, 2018, 17 (1) : 163-175. doi: 10.3934/cpaa.2018010
[15]	Reika Fukuizumi, Louis Jeanjean. Stability of standing waves for a nonlinear Schrödinger equation wdelta potentialith a repulsive Dirac. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 121-136. doi: 10.3934/dcds.2008.21.121
[16]	Jun-ichi Segata. Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves. Communications on Pure and Applied Analysis, 2015, 14 (3) : 843-859. doi: 10.3934/cpaa.2015.14.843
[17]	Yi He, Gongbao Li. Concentrating solitary waves for a class of singularly perturbed quasilinear Schrödinger equations with a general nonlinearity. Mathematical Control and Related Fields, 2016, 6 (4) : 551-593. doi: 10.3934/mcrf.2016016
[18]	Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328
[19]	Benedetta Noris, Hugo Tavares, Gianmaria Verzini. Stable solitary waves with prescribed $L^2$-mass for the cubic Schrödinger system with trapping potentials. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 6085-6112. doi: 10.3934/dcds.2015.35.6085
[20]	Xiaoyu Zeng. Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1749-1762. doi: 10.3934/dcds.2017073

2020 Impact Factor: 1.916

Tools

Metrics

Other articles
by authors

on AIMS
Alexander Komech Elena Kopylova David Stuart
on Google Scholar
Alexander Komech Elena Kopylova David Stuart

[Back to Top]