October  2012, 5(5): 971-987. doi: 10.3934/dcdss.2012.5.971

On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation

1. 

Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan

2. 

Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1

Received  March 2011 Revised  June 2011 Published  January 2012

Asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation was earlier established for septic and higher-order nonlinear terms by using Strichartz estimate. We use here pointwise dispersive decay estimates to push down the lower bound for the exponent of the nonlinear terms.
Citation: Tetsu Mizumachi, Dmitry Pelinovsky. On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 971-987. doi: 10.3934/dcdss.2012.5.971
References:
[1]

J. Belmonte-Beitia and D. Pelinovsky, Bifurcation of gap solitons in periodic potentials with a sign-varying nonlinearity coefficient,, Applic. Anal., 89 (2010), 1335. Google Scholar

[2]

V. S. Buslaev and G. S. Perelman, Scattering for the nonlinear Schrödinger equation: States close to a soliton,, St. Petersburg Math. J., 4 (1993), 1111. Google Scholar

[3]

S. Cuccagna, Orbitally but not asymptotically stable ground states for the discrete NLS,, Discrete Contin. Dyn. Syst., 26 (2010), 105. doi: 10.3934/dcds.2010.26.105. Google Scholar

[4]

S. Cuccagna and M. Tarulli, On asymptotic stability of standing waves of discrete Schrödinger equation in $\mathbbZ$,, SIAM J. Math. Anal., 41 (2009), 861. Google Scholar

[5]

P. G. Kevrekidis, D. E. Pelinovsky and A. Stefanov, Asymptotic stability of small bound states in the discrete nonlinear Schrödinger equation,, SIAM J. Math. Anal., 41 (2009), 2010. doi: 10.1137/080737654. Google Scholar

[6]

E. Kirr and Ö. Mizrak, Asymptotic stability of ground states in 3D nonlinear Schrödinger equation including subcritical cases,, J. Funct. Anal., 257 (2009), 3691. doi: 10.1016/j.jfa.2009.08.010. Google Scholar

[7]

E. Kirr and A. Zarnescu, Asymptotic stability of ground states in 2D nonlinear Schrödinger equation including subcritical cases,, J. Diff. Eqs., 247 (2009), 710. doi: 10.1016/j.jde.2009.04.015. Google Scholar

[8]

A. Komech, E. Kopylova and M. Kunze, Dispersive estimates for 1D discrete Schrödinger and Klein-Gordon equations,, Applic. Anal., 85 (2006), 1487. doi: 10.1080/00036810601074321. Google Scholar

[9]

J. Krieger and W. Schlag, Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension,, J. Amer. Math. Soc., 19 (2006), 815. doi: 10.1090/S0894-0347-06-00524-8. Google Scholar

[10]

F. Linares and G. Ponce, "Introduction to Nonlinear Dispersive Equations,", Universitext, (2009). Google Scholar

[11]

A. Mielke and C. Patz, Dispersive stability of infinite-dimensional Hamiltonian systems on lattices,, Applic. Anal., 89 (2010), 1493. doi: 10.1080/00036810903517605. Google Scholar

[12]

G. M. N'Guérékata and A. Pankov, Global well-posedness for discrete nonlinear Schrödinger equation,, Applic. Anal., 89 (2010), 1513. Google Scholar

[13]

P. Pacciani, V. V. Konotop and G. Perla Menzala, On localized solutions of discrete nonlinear Schrödinger equation. An exact result,, Physica D, 204 (2005), 122. doi: 10.1016/j.physd.2005.04.009. Google Scholar

[14]

A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations,, Nonlinearity, 19 (2006), 27. doi: 10.1088/0951-7715/19/1/002. Google Scholar

[15]

P. Panayotaros and D. Pelinovsky, Periodic oscillations of discrete NLS solitons in the presence of diffraction management,, Nonlinearity, 21 (2008), 1265. doi: 10.1088/0951-7715/21/6/007. Google Scholar

[16]

D. Pelinovsky and A. Stefanov, On the spectral theory and dispersive estimates for a discrete Schrödinger equation in one dimension,, J. Math. Phys., 49 (2008). Google Scholar

[17]

D. Pelinovsky and A. Sakovich, Internal modes of discrete solitons near the anti-continuum limit of the dNLS equation,, Physica D, 240 (2011), 265. doi: 10.1016/j.physd.2010.09.002. Google Scholar

[18]

W. Schlag, Dispersive estimates for Schrödinger operators: A survey,, in, 163 (2007), 255. Google Scholar

[19]

A. Stefanov and P. Kevrekidis, Asymptotic behaviour of small solutions for the discrete nonlinear Schrödinger and Klein-Gordon equations,, Nonlinearity, 18 (2005), 1841. doi: 10.1088/0951-7715/18/4/022. Google Scholar

[20]

M. I. Weinstein, Excitation thresholds for nonlinear localized modes on lattices,, Nonlinearity, 12 (1999), 673. doi: 10.1088/0951-7715/12/3/314. Google Scholar

show all references

References:
[1]

J. Belmonte-Beitia and D. Pelinovsky, Bifurcation of gap solitons in periodic potentials with a sign-varying nonlinearity coefficient,, Applic. Anal., 89 (2010), 1335. Google Scholar

[2]

V. S. Buslaev and G. S. Perelman, Scattering for the nonlinear Schrödinger equation: States close to a soliton,, St. Petersburg Math. J., 4 (1993), 1111. Google Scholar

[3]

S. Cuccagna, Orbitally but not asymptotically stable ground states for the discrete NLS,, Discrete Contin. Dyn. Syst., 26 (2010), 105. doi: 10.3934/dcds.2010.26.105. Google Scholar

[4]

S. Cuccagna and M. Tarulli, On asymptotic stability of standing waves of discrete Schrödinger equation in $\mathbbZ$,, SIAM J. Math. Anal., 41 (2009), 861. Google Scholar

[5]

P. G. Kevrekidis, D. E. Pelinovsky and A. Stefanov, Asymptotic stability of small bound states in the discrete nonlinear Schrödinger equation,, SIAM J. Math. Anal., 41 (2009), 2010. doi: 10.1137/080737654. Google Scholar

[6]

E. Kirr and Ö. Mizrak, Asymptotic stability of ground states in 3D nonlinear Schrödinger equation including subcritical cases,, J. Funct. Anal., 257 (2009), 3691. doi: 10.1016/j.jfa.2009.08.010. Google Scholar

[7]

E. Kirr and A. Zarnescu, Asymptotic stability of ground states in 2D nonlinear Schrödinger equation including subcritical cases,, J. Diff. Eqs., 247 (2009), 710. doi: 10.1016/j.jde.2009.04.015. Google Scholar

[8]

A. Komech, E. Kopylova and M. Kunze, Dispersive estimates for 1D discrete Schrödinger and Klein-Gordon equations,, Applic. Anal., 85 (2006), 1487. doi: 10.1080/00036810601074321. Google Scholar

[9]

J. Krieger and W. Schlag, Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension,, J. Amer. Math. Soc., 19 (2006), 815. doi: 10.1090/S0894-0347-06-00524-8. Google Scholar

[10]

F. Linares and G. Ponce, "Introduction to Nonlinear Dispersive Equations,", Universitext, (2009). Google Scholar

[11]

A. Mielke and C. Patz, Dispersive stability of infinite-dimensional Hamiltonian systems on lattices,, Applic. Anal., 89 (2010), 1493. doi: 10.1080/00036810903517605. Google Scholar

[12]

G. M. N'Guérékata and A. Pankov, Global well-posedness for discrete nonlinear Schrödinger equation,, Applic. Anal., 89 (2010), 1513. Google Scholar

[13]

P. Pacciani, V. V. Konotop and G. Perla Menzala, On localized solutions of discrete nonlinear Schrödinger equation. An exact result,, Physica D, 204 (2005), 122. doi: 10.1016/j.physd.2005.04.009. Google Scholar

[14]

A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations,, Nonlinearity, 19 (2006), 27. doi: 10.1088/0951-7715/19/1/002. Google Scholar

[15]

P. Panayotaros and D. Pelinovsky, Periodic oscillations of discrete NLS solitons in the presence of diffraction management,, Nonlinearity, 21 (2008), 1265. doi: 10.1088/0951-7715/21/6/007. Google Scholar

[16]

D. Pelinovsky and A. Stefanov, On the spectral theory and dispersive estimates for a discrete Schrödinger equation in one dimension,, J. Math. Phys., 49 (2008). Google Scholar

[17]

D. Pelinovsky and A. Sakovich, Internal modes of discrete solitons near the anti-continuum limit of the dNLS equation,, Physica D, 240 (2011), 265. doi: 10.1016/j.physd.2010.09.002. Google Scholar

[18]

W. Schlag, Dispersive estimates for Schrödinger operators: A survey,, in, 163 (2007), 255. Google Scholar

[19]

A. Stefanov and P. Kevrekidis, Asymptotic behaviour of small solutions for the discrete nonlinear Schrödinger and Klein-Gordon equations,, Nonlinearity, 18 (2005), 1841. doi: 10.1088/0951-7715/18/4/022. Google Scholar

[20]

M. I. Weinstein, Excitation thresholds for nonlinear localized modes on lattices,, Nonlinearity, 12 (1999), 673. doi: 10.1088/0951-7715/12/3/314. Google Scholar

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