On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D
Perelman, Galina1
1 Centre de Mathématiques, Ecole Polytechnique, F-91128 Palaiseau Cedex
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1999-2000), Exposé no. 3, 14 p.
Perelman, Galina 1

1 Centre de Mathématiques, Ecole Polytechnique, F-91128 Palaiseau Cedex 
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     author = {Perelman, Galina},
     title = {On the blow up phenomenon for the critical nonlinear {Schr\"odinger} equation in {1D}},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
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     pages = {1--14},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {1999-2000},
     zbl = {1061.35523},
     mrnumber = {1813166},
     language = {en},
     url = {http://archive.numdam.org/item/SEDP_1999-2000____A3_0/}
}
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Perelman, Galina. On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1999-2000), Exposé no. 3, 14 p. http://archive.numdam.org/item/SEDP_1999-2000____A3_0/

[BW] 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. | Numdam | MR | Zbl

[BP] 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. | MR | Zbl

[CW] 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. | MR | Zbl

[Gl] R.Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger operators, J. Math. Phys. 8 (1977), 1794-1797. | MR | Zbl

[GV1] J.Ginibre, G.Velo, On a class of nonlinear Schrödinger equations I, II, J. Func. Anal. 32 (1979), 1-71. | MR | Zbl

[GV2] J.Ginibre, G.Velo, On a class of nonlinear Schrödinger equations III, Ann. Inst. H. Poincaré Phys. Thér. 28 (1978), 287-316. | Numdam | MR | Zbl

[Gr] M.Grillakis, Linearized instability for nonlinear Schrödinger and Klein - Gordon equations, Comm. Pure Appl. Math. (1988), 747-774. | MR | Zbl

[KSZ] N.Kosmatov, V.Schvets, V.Zakharov, Computer simulation of wave collapse in nonlinear Schrödinger equation,Phys. D 52 (1991), 16-35. | Zbl

[LBSK] 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. | MR | Zbl

[LPSS1] 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. | MR | Zbl

[LPSS2] 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. | MR | Zbl

[Ma] V.M.Malkin, Dynamics of wave collapse in the critical case, Phys. Lett. A 151 (1990), 285-288.

[Me] F. Merle, Determination of blow- up solutions with minimal mass for nonlinear Schrödinger equation with critical power, Duke Math. J. 69 (1993), 427-453. | MR | Zbl

[P] G.Perelman, On the formation of singularities in solutions of nonlinear Schrödinger equation with critical power nonlinearity, in preparation.

[SF] A.I.Smirnov, G.M.Fraiman, The interaction representation in the self-focusing theory, Phys. D 51 (1991), 2-15. | MR | Zbl

[SW1] A.Soffer and M.I.Weinstein, Multichannel nonlinear scattering theory for nonintegrable equations I, Comm. Math. Phys. 133 (1990), 119-146. | MR | Zbl

[SW2] A.Soffer and M.I.Weinstein, Multichannel nonlinear scattering theory for nonintegrable equations II, J. Diff. Eq. 98 (1992), 376-390. | MR | Zbl

[SS] C.Sulem, P.Sulem, Focusing nonlinear Schrödinger equation and wave-packet collapse, Nonlin. Anal., Theory, Meth. & Appl. 30(2) (1997), 833-844. | MR | Zbl

[W1] M.I.Weinstein, Modulation stability of ground states of nonlinear Schrödinger equation, SIAM J. Math. Anal. 16 (1985), 472-491. | MR | Zbl

[W2] M.I.Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Commun. Pure Appl. Math. 39 (1986), 51-68. | MR | Zbl

[W3] M.I.Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983), 567-576. | MR | Zbl

[W4] 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. | MR | Zbl

[W5] M.I.Weinstein, Solitary waves of nonlinear dispersive evolution equations with critical power nonlinearities, J. Diff. Eq. 69 (1987), 192-203. | MR | Zbl