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Existence and regularity of solutions in nonlinear wave equations
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 |
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. |
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), 197-257. |
[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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