On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D

Galina Perelman[1]

  • [1] Centre de Mathématiques, Ecole Polytechnique, F-91128 Palaiseau Cedex

Séminaire Équations aux dérivées partielles (1999-2000)

  • Volume: 1999-2000, page 1-14

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Perelman, Galina. "On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D." Séminaire Équations aux dérivées partielles 1999-2000 (1999-2000): 1-14. <http://eudml.org/doc/11000>.

@article{Perelman1999-2000,
affiliation = {Centre de Mathématiques, Ecole Polytechnique, F-91128 Palaiseau Cedex},
author = {Perelman, Galina},
journal = {Séminaire Équations aux dérivées partielles},
keywords = {ground state solitary wave; initial perturbations; global existence; blow up phenomenon; singularity formation},
language = {eng},
pages = {1-14},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D},
url = {http://eudml.org/doc/11000},
volume = {1999-2000},
year = {1999-2000},
}

TY - JOUR
AU - Perelman, Galina
TI - On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D
JO - Séminaire Équations aux dérivées partielles
PY - 1999-2000
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 1999-2000
SP - 1
EP - 14
LA - eng
KW - ground state solitary wave; initial perturbations; global existence; blow up phenomenon; singularity formation
UR - http://eudml.org/doc/11000
ER -

References

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  1. J.Bourgain, W.Wang, Construction of blow- up solutions for the nonlinear Schrödinger equation with critical nonlinearity, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4) XXV (1997), 197-215. Zbl1043.35137MR1655515
  2. V.S.Buslaev, G.S.Perelman, Scattering for the nonlinear Schrödinger equation: states close to a soliton, St. Peters. Math. J. 4 (1993), 1111-1143. Zbl0853.35112MR1199635
  3. T.Cazenave, F.B.Weissler, The structure of solutions to the pseudo-conformal invariant nonlinear Schrödinger equation, Proc. Royal Soc. Edinburgh 117A (1991), 251-273. Zbl0733.35094MR1103294
  4. R.Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger operators, J. Math. Phys. 8 (1977), 1794-1797. Zbl0372.35009MR460850
  5. J.Ginibre, G.Velo, On a class of nonlinear Schrödinger equations I, II, J. Func. Anal. 32 (1979), 1-71. Zbl0396.35029MR533219
  6. J.Ginibre, G.Velo, On a class of nonlinear Schrödinger equations III, Ann. Inst. H. Poincaré Phys. Thér. 28 (1978), 287-316. Zbl0397.35012MR498408
  7. M.Grillakis, Linearized instability for nonlinear Schrödinger and Klein - Gordon equations, Comm. Pure Appl. Math. (1988), 747-774. Zbl0632.70015MR948770
  8. N.Kosmatov, V.Schvets, V.Zakharov, Computer simulation of wave collapse in nonlinear Schrödinger equation,Phys. D 52 (1991), 16-35. Zbl0735.76085
  9. E.W.Laedke, R.Blaha, K.H.Spatschek, E.A.Kuznetsov, On the stability of collapse in the critical case, J. Math. Phys. 33(3) (1992), 967-973. Zbl0765.35053MR1149217
  10. B.J.LeMesurier, G.Papanicolau, C.Sulem, P.Sulem, Focusing and multi-focusing solutions of the nonlinear Schrödinger equation, Phys. D 31 (1988), 78-102. Zbl0694.35196MR947896
  11. B.J.LeMesurier, G.Papanicolau, C.Sulem, P.Sulem, Local structure of the self-focusing singularity of the nonlinear Schrödinger equation, Phys. D 32 (1988), 210-226. Zbl0694.35206MR969030
  12. V.M.Malkin, Dynamics of wave collapse in the critical case, Phys. Lett. A 151 (1990), 285-288. 
  13. F. Merle, Determination of blow- up solutions with minimal mass for nonlinear Schrödinger equation with critical power, Duke Math. J. 69 (1993), 427-453. Zbl0808.35141MR1203233
  14. G.Perelman, On the formation of singularities in solutions of nonlinear Schrödinger equation with critical power nonlinearity, in preparation. Zbl1007.35087
  15. A.I.Smirnov, G.M.Fraiman, The interaction representation in the self-focusing theory, Phys. D 51 (1991), 2-15. Zbl0735.76092MR1131136
  16. A.Soffer and M.I.Weinstein, Multichannel nonlinear scattering theory for nonintegrable equations I, Comm. Math. Phys. 133 (1990), 119-146. Zbl0721.35082MR1071238
  17. A.Soffer and M.I.Weinstein, Multichannel nonlinear scattering theory for nonintegrable equations II, J. Diff. Eq. 98 (1992), 376-390. Zbl0795.35073MR1170476
  18. C.Sulem, P.Sulem, Focusing nonlinear Schrödinger equation and wave-packet collapse, Nonlin. Anal., Theory, Meth. & Appl. 30(2) (1997), 833-844. Zbl0886.35140MR1487664
  19. M.I.Weinstein, Modulation stability of ground states of nonlinear Schrödinger equation, SIAM J. Math. Anal. 16 (1985), 472-491. Zbl0583.35028MR783974
  20. M.I.Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Commun. Pure Appl. Math. 39 (1986), 51-68. Zbl0594.35005MR820338
  21. M.I.Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983), 567-576. Zbl0527.35023MR691044
  22. M.I.Weinstein, On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations, Comm. Part. Diff. Eq. 11(5) (1986), 545-565. Zbl0596.35022MR829596
  23. M.I.Weinstein, Solitary waves of nonlinear dispersive evolution equations with critical power nonlinearities, J. Diff. Eq. 69 (1987), 192-203. Zbl0636.35015MR899159

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