2015, 35(8): 3343-3376. doi: 10.3934/dcds.2015.35.3343

On weak interaction between a ground state and a trapping potential

1. 

Department of Mathematics and Geosciences, University of Trieste, via Valerio 12/1 Trieste, 34127, Italy

2. 

Department of Mathematics and Informatics, Faculty of Science, Chiba University, Chiba 263-8522, Japan

Received  April 2014 Revised  December 2014 Published  February 2015

We continue our study initiated in [4] of the interaction of a ground state with a potential considering here a class of trapping potentials. We track the precise asymptotic behavior of the solution if the interaction is weak, either because the ground state moves away from the potential or is very fast.
Citation: Scipio Cuccagna, Masaya Maeda. On weak interaction between a ground state and a trapping potential. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3343-3376. doi: 10.3934/dcds.2015.35.3343
References:
[1]

S. Cuccagna, On the Darboux and Birkhoff steps in the asymptotic stability of solitons,, Rend. Istit. Mat. Univ. Trieste, 44 (2012), 197.

[2]

S. Cuccagna, The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states,, Comm. Math. Physics, 305 (2011), 279. doi: 10.1007/s00220-011-1265-2.

[3]

S. Cuccagna, On asymptotic stability of moving ground states of the nonlinear Schrödinger equation,, Trans. Amer. Math. Soc., 366 (2014), 2827. doi: 10.1090/S0002-9947-2014-05770-X.

[4]

S. Cuccagna and M. Maeda, On weak interaction between a ground state and a non-trapping potential,, J. Differential Equations, 256 (2014), 1395. doi: 10.1016/j.jde.2013.11.002.

[5]

S. Cuccagna and M. Maeda, On small energy stabilization in the NLS with a trapping potential,, preprint, ().

[6]

S. Cuccagna, D. Pelinovsky and V. Vougalter, Spectra of positive and negative energies in the linearization of the NLS problem,, Comm. Pure Appl. Math., 58 (2005), 1. doi: 10.1002/cpa.20050.

[7]

K. Datchev and J. Holmer, Fast soliton scattering by attractive delta impurities,, Comm. Partial Differential Equations, 34 (2009), 1074. doi: 10.1080/03605300903076831.

[8]

P. Deift and X. Zhou, Perturbation theory for infinite-dimensional integrable systems on the line. A case study,, Acta Math., 188 (2002), 163. doi: 10.1007/BF02392683.

[9]

M. Grillakis, J. Shatah and W. Strauss, Stability of solitary waves in the presence of symmetries, I,, Jour. Funct. An., 74 (1987), 160. doi: 10.1016/0022-1236(87)90044-9.

[10]

S. Gustafson, K. Nakanishi and T. P. Tsai, Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves,, Int. Math. Res. Not., 66 (2004), 3559. doi: 10.1155/S1073792804132340.

[11]

J. Holmer, J. Marzuola and M. Zworski, Fast soliton scattering by delta impurities,, Comm. Math. Physics, 274 (2007), 187. doi: 10.1007/s00220-007-0261-z.

[12]

J. Holmer, J. Marzuola and M. Zworski, Soliton splitting by external delta potentials,, J. Nonlinear Sci., 17 (2007), 349. doi: 10.1007/s00332-006-0807-9.

[13]

Y. Martel and F. Merle, Review of long time asymptotics and collision of solitons for the quartic generalized Korteweg-de Vries equation,, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 287. doi: 10.1017/S030821051000003X.

[14]

Y. Martel, F. Merle and T. P. Tsai, Stability in $H^1$ of the sum of K solitary waves for some nonlinear Schrödinger equations,, Duke Math. J., 133 (2006), 405. doi: 10.1215/S0012-7094-06-13331-8.

[15]

G. Perelman, Two soliton collision for nonlinear Schrödinger equations in dimension 1,, Ann. Inst. H. Poinc. Anal. Non Lin., 28 (2011), 357. doi: 10.1016/j.anihpc.2011.02.002.

[16]

G. Perelman, A remark on soliton-potential interactions for nonlinear Schrödinger equations,, Math. Res. Lett., 16 (2009), 477. doi: 10.4310/MRL.2009.v16.n3.a8.

[17]

G. Perelman, Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations,, Comm. Partial Diff., 29 (2004), 1051. doi: 10.1081/PDE-200033754.

[18]

I. Rodnianski, W. Schlag and A. Soffer, Dispersive analysis of charge transfer models,, Comm. Pure Appl. Math., 58 (2005), 149. doi: 10.1002/cpa.20066.

[19]

I. Rodnianski, W. Schlag and A. Soffer, Asymptotic stability of N-soliton states of NLS,, preprint, ().

[20]

M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive equations,, Comm. Pure Appl. Math., 39 (1986), 51. doi: 10.1002/cpa.3160390103.

show all references

References:
[1]

S. Cuccagna, On the Darboux and Birkhoff steps in the asymptotic stability of solitons,, Rend. Istit. Mat. Univ. Trieste, 44 (2012), 197.

[2]

S. Cuccagna, The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states,, Comm. Math. Physics, 305 (2011), 279. doi: 10.1007/s00220-011-1265-2.

[3]

S. Cuccagna, On asymptotic stability of moving ground states of the nonlinear Schrödinger equation,, Trans. Amer. Math. Soc., 366 (2014), 2827. doi: 10.1090/S0002-9947-2014-05770-X.

[4]

S. Cuccagna and M. Maeda, On weak interaction between a ground state and a non-trapping potential,, J. Differential Equations, 256 (2014), 1395. doi: 10.1016/j.jde.2013.11.002.

[5]

S. Cuccagna and M. Maeda, On small energy stabilization in the NLS with a trapping potential,, preprint, ().

[6]

S. Cuccagna, D. Pelinovsky and V. Vougalter, Spectra of positive and negative energies in the linearization of the NLS problem,, Comm. Pure Appl. Math., 58 (2005), 1. doi: 10.1002/cpa.20050.

[7]

K. Datchev and J. Holmer, Fast soliton scattering by attractive delta impurities,, Comm. Partial Differential Equations, 34 (2009), 1074. doi: 10.1080/03605300903076831.

[8]

P. Deift and X. Zhou, Perturbation theory for infinite-dimensional integrable systems on the line. A case study,, Acta Math., 188 (2002), 163. doi: 10.1007/BF02392683.

[9]

M. Grillakis, J. Shatah and W. Strauss, Stability of solitary waves in the presence of symmetries, I,, Jour. Funct. An., 74 (1987), 160. doi: 10.1016/0022-1236(87)90044-9.

[10]

S. Gustafson, K. Nakanishi and T. P. Tsai, Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves,, Int. Math. Res. Not., 66 (2004), 3559. doi: 10.1155/S1073792804132340.

[11]

J. Holmer, J. Marzuola and M. Zworski, Fast soliton scattering by delta impurities,, Comm. Math. Physics, 274 (2007), 187. doi: 10.1007/s00220-007-0261-z.

[12]

J. Holmer, J. Marzuola and M. Zworski, Soliton splitting by external delta potentials,, J. Nonlinear Sci., 17 (2007), 349. doi: 10.1007/s00332-006-0807-9.

[13]

Y. Martel and F. Merle, Review of long time asymptotics and collision of solitons for the quartic generalized Korteweg-de Vries equation,, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 287. doi: 10.1017/S030821051000003X.

[14]

Y. Martel, F. Merle and T. P. Tsai, Stability in $H^1$ of the sum of K solitary waves for some nonlinear Schrödinger equations,, Duke Math. J., 133 (2006), 405. doi: 10.1215/S0012-7094-06-13331-8.

[15]

G. Perelman, Two soliton collision for nonlinear Schrödinger equations in dimension 1,, Ann. Inst. H. Poinc. Anal. Non Lin., 28 (2011), 357. doi: 10.1016/j.anihpc.2011.02.002.

[16]

G. Perelman, A remark on soliton-potential interactions for nonlinear Schrödinger equations,, Math. Res. Lett., 16 (2009), 477. doi: 10.4310/MRL.2009.v16.n3.a8.

[17]

G. Perelman, Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations,, Comm. Partial Diff., 29 (2004), 1051. doi: 10.1081/PDE-200033754.

[18]

I. Rodnianski, W. Schlag and A. Soffer, Dispersive analysis of charge transfer models,, Comm. Pure Appl. Math., 58 (2005), 149. doi: 10.1002/cpa.20066.

[19]

I. Rodnianski, W. Schlag and A. Soffer, Asymptotic stability of N-soliton states of NLS,, preprint, ().

[20]

M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive equations,, Comm. Pure Appl. Math., 39 (1986), 51. doi: 10.1002/cpa.3160390103.

[1]

Alex H. Ardila. Stability of ground states for logarithmic Schrödinger equation with a $\delta^{'}$-interaction. Evolution Equations & Control Theory, 2017, 6 (2) : 155-175. doi: 10.3934/eect.2017009

[2]

Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395

[3]

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

[4]

Chuangye Liu, Zhi-Qiang Wang. A complete classification of ground-states for a coupled nonlinear Schrödinger system. Communications on Pure & Applied Analysis, 2017, 16 (1) : 115-130. doi: 10.3934/cpaa.2017005

[5]

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

[6]

Eugenio Montefusco, Benedetta Pellacci, Marco Squassina. Energy convexity estimates for non-degenerate ground states of nonlinear 1D Schrödinger systems. Communications on Pure & Applied Analysis, 2010, 9 (4) : 867-884. doi: 10.3934/cpaa.2010.9.867

[7]

Chang-Lin Xiang. Remarks on nondegeneracy of ground states for quasilinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5789-5800. doi: 10.3934/dcds.2016054

[8]

Daniele Garrisi, Vladimir Georgiev. Orbital stability and uniqueness of the ground state for the non-linear Schrödinger equation in dimension one. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4309-4328. doi: 10.3934/dcds.2017184

[9]

Chenmin Sun, Hua Wang, Xiaohua Yao, Jiqiang Zheng. Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2207-2228. doi: 10.3934/dcds.2018091

[10]

Kazuhiro Kurata, Tatsuya Watanabe. A remark on asymptotic profiles of radial solutions with a vortex to a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 597-610. doi: 10.3934/cpaa.2006.5.597

[11]

Nakao Hayashi, Pavel I. Naumkin. Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 383-400. doi: 10.3934/dcds.1997.3.383

[12]

Reika Fukuizumi. Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 525-544. doi: 10.3934/dcds.2001.7.525

[13]

François Genoud. Existence and stability of high frequency standing waves for a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1229-1247. doi: 10.3934/dcds.2009.25.1229

[14]

Silvia Cingolani, Mónica Clapp. Symmetric semiclassical states to a magnetic nonlinear Schrödinger equation via equivariant Morse theory. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1263-1281. doi: 10.3934/cpaa.2010.9.1263

[15]

Giuseppe Maria Coclite, Helge Holden. Ground states of the Schrödinger-Maxwell system with dirac mass: Existence and asymptotics. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 117-132. doi: 10.3934/dcds.2010.27.117

[16]

D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563

[17]

Tetsu Mizumachi. Instability of bound states for 2D nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 413-428. doi: 10.3934/dcds.2005.13.413

[18]

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 & Applied Analysis, 2015, 14 (3) : 843-859. doi: 10.3934/cpaa.2015.14.843

[19]

Reika Fukuizumi, Louis Jeanjean. Stability of standing waves for a nonlinear Schrödinger equation wdelta potentialith a repulsive Dirac. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 121-136. doi: 10.3934/dcds.2008.21.121

[20]

Alex H. Ardila. Stability of standing waves for a nonlinear SchrÖdinger equation under an external magnetic field. Communications on Pure & Applied Analysis, 2018, 17 (1) : 163-175. doi: 10.3934/cpaa.2018010

2016 Impact Factor: 1.099

Metrics

  • PDF downloads (0)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]