American Institute of Mathematical Sciences

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

On weak interaction between a ground state and a trapping potential

Scipio Cuccagna 1, and  Masaya Maeda 2,

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.
Keywords: ground states, Nonlinear Schrödinger equation, asymptotic stability..
Mathematics Subject Classification: Primary: 35Q5.
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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[1]

Alex H. Ardila. Stability of ground states for logarithmic Schrödinger equation with a $δ^{\prime}$-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]

Zupei Shen, Zhiqing Han, Qinqin Zhang. Ground states of nonlinear Schrödinger equations with fractional Laplacians. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2115-2125. doi: 10.3934/dcdss.2019136
[6]

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
[7]

Wen Feng, Milena Stanislavova, Atanas Stefanov. On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1371-1385. doi: 10.3934/cpaa.2018067
[8]

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
[9]

Dongdong Qin, Xianhua Tang, Qingfang Wu. Ground states of nonlinear Schrödinger systems with periodic or non-periodic potentials. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1261-1280. doi: 10.3934/cpaa.2019061
[10]

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
[11]

Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074
[12]

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
[13]

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
[14]

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
[15]

Xiaoyu Zeng, Yimin Zhang. Asymptotic behaviors of ground states for a modified Gross-Pitaevskii equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5263-5273. doi: 10.3934/dcds.2019214
[16]

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
[17]

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
[18]

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
[19]

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
[20]

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

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences