On asymptotic stability in energy space of ground states of NLS in 2D

Scipio Cuccagna; Mirko Tarulli

Annales de l'I.H.P. Analyse non linéaire (2009)

  • Volume: 26, Issue: 4, page 1361-1386
  • ISSN: 0294-1449

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Cuccagna, Scipio, and Tarulli, Mirko. "On asymptotic stability in energy space of ground states of NLS in 2D." Annales de l'I.H.P. Analyse non linéaire 26.4 (2009): 1361-1386. <http://eudml.org/doc/78894>.

@article{Cuccagna2009,
author = {Cuccagna, Scipio, Tarulli, Mirko},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {nonlinear Schrödinger equation; asymptotic stability; ground states of NLS; wave operators; Fourier integral operators; Fermi golden rule (FGR); Strichartz estimates; smoothing estimates},
language = {eng},
number = {4},
pages = {1361-1386},
publisher = {Elsevier},
title = {On asymptotic stability in energy space of ground states of NLS in 2D},
url = {http://eudml.org/doc/78894},
volume = {26},
year = {2009},
}

TY - JOUR
AU - Cuccagna, Scipio
AU - Tarulli, Mirko
TI - On asymptotic stability in energy space of ground states of NLS in 2D
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2009
PB - Elsevier
VL - 26
IS - 4
SP - 1361
EP - 1386
LA - eng
KW - nonlinear Schrödinger equation; asymptotic stability; ground states of NLS; wave operators; Fourier integral operators; Fermi golden rule (FGR); Strichartz estimates; smoothing estimates
UR - http://eudml.org/doc/78894
ER -

References

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  1. [1] Agmon S., Spectral properties of Schrodinger operators and scattering theory, Ann. Sc. Norm. Pisa2 (1975) 151-218. Zbl0315.47007MR397194
  2. [2] Burq N., Global Strichartz estimates for nontrapping geometries: about an article by H. Smith and C. Sogge, Comm. Partial Differential Equations28 (2003) 1675-1683. Zbl1026.35020MR2001179
  3. [3] Buslaev V.S., Perelman G.S., Scattering for the nonlinear Schrödinger equation: states close to a soliton, St. Petersburg Math. J.4 (1993) 1111-1142. Zbl0853.35112MR1199635
  4. [4] Buslaev V.S., Perelman G.S., On the stability of solitary waves for nonlinear Schrödinger equations, in: Uraltseva N.N. (Ed.), Nonlinear Evolution Equations, Transl. Ser. 2, vol. 164, Amer. Math. Soc., Providence, RI, 1995, pp. 75-98. Zbl0841.35108MR1334139
  5. [5] Buslaev V.S., Sulem C., On the asymptotic stability of solitary waves of Nonlinear Schrödinger equations, Ann. Inst. H. Poincaré. Anal. Non Linéaire20 (2003) 419-475. Zbl1028.35139MR1972870
  6. [6] Christ M., Kieslev A., Maximal functions associated with filtrations, J. Funct. Anal.179 (2001) 409-425. Zbl0974.47025MR1809116
  7. [7] Cuccagna S., A revision of “On asymptotic stability in energy space of ground states of NLS in 1D”, http://arxiv.org/abs/0711.4192. MR2422523
  8. [8] Cuccagna S., Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure Appl. Math.54 (2001) 1110-1145. Zbl1031.35129MR1835384
  9. [9] Cuccagna S., On asymptotic stability of ground states of NLS, Rev. Math. Phys.15 (2003) 877-903. Zbl1084.35089MR2027616
  10. [10] Cuccagna S., Mizumachi T., On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations, Comm. Math. Phys.284 (2008) 51-77. Zbl1155.35092MR2443298
  11. [11] Cuccagna S., Pelinovsky D., Vougalter V., Spectra of positive and negative energies in the linearization of the NLS problem, Comm. Pure Appl. Math.58 (2005) 1-29. Zbl1064.35181MR2094265
  12. [12] Gang Z., Sigal I.M., Relaxation of solitons in nonlinear Schrödinger equations with potential, Adv. Math.216 (2007) 443-490. Zbl1126.35065MR2351368
  13. [13] Grillakis M., Shatah J., Strauss W., Stability of solitary waves in the presence of symmetries, I, J. Funct. Anal.74 (1987) 160-197. Zbl0656.35122MR901236
  14. [14] Grillakis M., Shatah J., Strauss W., Stability of solitary waves in the presence of symmetries, II, J. Funct. Anal.94 (1990) 308-348. Zbl0711.58013MR1081647
  15. [15] Gustafson S., Nakanishi K., Tsai T.P., Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves, Int. Math. Res. Notices66 (2004) 3559-3584. Zbl1072.35167MR2101699
  16. [16] Jensen A., Kato T., Spectral properties of Schrödinger operators and time decay of the wave functions, Duke Math. J.46 (1979) 583-611. Zbl0448.35080MR544248
  17. [17] Jensen A., Nenciu G., A unified approach to resolvent expansions at thresholds, Rev. Math. Phys.13 (2001) 717-754. Zbl1029.81067MR1841744
  18. [18] Jensen A., Yajima K., A remark on L p boundedness of wave operators for two-dimensional Schrödinger operators, Comm. Math. Phys.225 (2002) 633-637. Zbl1057.47011MR1888876
  19. [19] Kato T., Wave operators and similarity for some non-selfadjoint operators, Math. Ann.162 (1966) 258-269. Zbl0139.31203MR190801
  20. [20] Kirr E., Zarnescu A., On the asymptotic stability of bound states in 2D cubic Scrödinger equation, Comm. Math. Phys.272 (2007) 443-468. Zbl1194.35416MR2300249
  21. [21] Mizumachi T., Asymptotic stability of small solitons to 1D NLS with potential, http://arxiv.org/abs/math.AP/0605031. 
  22. [22] Mizumachi T., Asymptotic stability of small solitons for 2D nonlinear Schrödinger equations with potential, J. Math. Kyoto Univ.47 (2007) 599-620. Zbl1146.35085MR2402517
  23. [23] Perelman G.S., On the formation of singularities in solutions of the critical nonlinear Schrödinger equation, Ann. Henri Poincaré2 (2001) 605-673. Zbl1007.35087MR1852922
  24. [24] Pillet C.A., Wayne C.E., Invariant manifolds for a class of dispersive, Hamiltonian partial differential equations, J. Diff. Eq.141 (1997) 310-326. Zbl0890.35016MR1488355
  25. [25] Reed M., Simon B., Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, 1978. Zbl0242.46001MR751959
  26. [26] Schlag W., Dispersive estimates for Schrödinger operators in dimension two, Comm. Math. Phys.257 (2005) 87-117. Zbl1134.35321MR2163570
  27. [27] Shatah J., Strauss W., Instability of nonlinear bound states, Comm. Math. Phys.100 (1985) 173-190. Zbl0603.35007MR804458
  28. [28] Smith H.F., Sogge C.D., Global Strichartz estimates for nontrapping perturbations of the Laplacian, Comm. Partial Differential Equations25 (2000) 2171-2183. Zbl0972.35014MR1789924
  29. [29] Soffer A., Weinstein M., Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys.133 (1990) 116-146. Zbl0721.35082MR1071238
  30. [30] Soffer A., Weinstein M., Multichannel nonlinear scattering II. The case of anisotropic potentials and data, J. Differential Equations98 (1992) 376-390. Zbl0795.35073MR1170476
  31. [31] Soffer A., Weinstein M., Selection of the ground state for nonlinear Schrödinger equations, Rev. Math. Phys.16 (2004) 977-1071. Zbl1111.81313MR2101776
  32. [32] Taylor M.E., Partial Differential Equations II, Appl. Math. Sci., vol. 116, Springer, 1997. MR1395149
  33. [33] Tsai T.P., Yau H.T., Asymptotic dynamics of nonlinear Schrödinger equations: resonance dominated and radiation dominated solutions, Comm. Pure Appl. Math.55 (2002) 153-216. Zbl1031.35137MR1865414
  34. [34] Tsai T.P., Yau H.T., Relaxation of excited states in nonlinear Schrödinger equations, Int. Math. Res. Notices31 (2002) 1629-1673. Zbl1011.35120MR1916427
  35. [35] Tsai T.P., Yau H.T., Classification of asymptotic profiles for nonlinear Schrödinger equations with small initial data, Adv. Theor. Math. Phys.6 (2002) 107-139. Zbl1033.81034MR1992875
  36. [36] Weder R., Center manifold for nonintegrable nonlinear Schrödinger equations on the line, Comm. Math. Phys.170 (2000) 343-356. Zbl1003.37045MR1799850
  37. [37] Weinstein M., Modulation stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal.16 (1985) 472-491. Zbl0583.35028MR783974
  38. [38] Weinstein M., Lyapunov stability of ground states of nonlinear dispersive equations, Comm. Pure Appl. Math.39 (1986) 51-68. Zbl0594.35005MR820338
  39. [39] Yajima K., The W k , p continuity of wave operators for Schrödinger operators, J. Math. Soc. Japan47 (1995) 551-581. Zbl0837.35039MR1331331
  40. [40] Yajima K., The L p boundedness of wave operators for two dimensional Schrödinger operators, Comm. Math. Phys.208 (1999) 125-152. Zbl0961.47004MR1729881

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