# American Institute of Mathematical Sciences

August  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 and Continuous Dynamical Systems, 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), 197-257. [2] S. Cuccagna, The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states, Comm. Math. Physics, 305 (2011), 279-331. 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), 2827-2888. 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), 1395-1466. 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, arXiv:1309.4878. [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), 1-29. doi: 10.1002/cpa.20050. [7] K. Datchev and J. Holmer, Fast soliton scattering by attractive delta impurities, Comm. Partial Differential Equations, 34 (2009), 1074-1113. 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), 163-262. 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), 160-197. 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), 3559-3584. doi: 10.1155/S1073792804132340. [11] J. Holmer, J. Marzuola and M. Zworski, Fast soliton scattering by delta impurities, Comm. Math. Physics, 274 (2007), 187-216. 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), 349-367. 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), 287-317. 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), 405-466. 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), 357-384. 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), 477-486. 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), 1051-1095. 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), 149-216. doi: 10.1002/cpa.20066. [19] I. Rodnianski, W. Schlag and A. Soffer, Asymptotic stability of N-soliton states of NLS, preprint, arXiv:math/0309114. [20] M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive equations, Comm. Pure Appl. Math., 39 (1986), 51-67. doi: 10.1002/cpa.3160390103.

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##### References:
 [1] S. Cuccagna, On the Darboux and Birkhoff steps in the asymptotic stability of solitons, Rend. Istit. Mat. Univ. Trieste, 44 (2012), 197-257. [2] S. Cuccagna, The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states, Comm. Math. Physics, 305 (2011), 279-331. 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), 2827-2888. 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), 1395-1466. 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, arXiv:1309.4878. [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), 1-29. doi: 10.1002/cpa.20050. [7] K. Datchev and J. Holmer, Fast soliton scattering by attractive delta impurities, Comm. Partial Differential Equations, 34 (2009), 1074-1113. 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), 163-262. 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), 160-197. 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), 3559-3584. doi: 10.1155/S1073792804132340. [11] J. Holmer, J. Marzuola and M. Zworski, Fast soliton scattering by delta impurities, Comm. Math. Physics, 274 (2007), 187-216. 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), 349-367. 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), 287-317. 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), 405-466. 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), 357-384. 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), 477-486. 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), 1051-1095. 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), 149-216. doi: 10.1002/cpa.20066. [19] I. Rodnianski, W. Schlag and A. Soffer, Asymptotic stability of N-soliton states of NLS, preprint, arXiv:math/0309114. [20] M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive equations, Comm. Pure Appl. Math., 39 (1986), 51-67. doi: 10.1002/cpa.3160390103.
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