Non-generic blow-up solutions for the critical focusing NLS in 1-D

Journal of the European Mathematical Society

Volumn 11, Issue 1, 2009, Pages 1-125

  Krieger, J ^a   Schlag, W ^b  

^a HARVARD UNIVERSITY   (United States)

^b UNIVERSITY OF CHICAGO   (United States)

Author keywords

L2 critical NLS; Non linear Schr dinger equations; Pseudo conformal blow up

Indexed keywords

EID: 70350005122     PISSN: 14359855     EISSN: None     Source Type: Journal    
DOI: 10.4171/JEMS/143     Document Type: Article

References (32)

1. [Agm] Agmon, S.: Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-body Schrödinger Operators. Math. Notes 29, Princeton Univ. Press, Princeton, NJ, University of Tokyo Press, Tokyo (1982) Zbl 0503.35001 MR 0745286
2. [BerCaz] Berestycki, H., Cazenave, T.: Instabilité des états stationnaires dans les équations deSchrödinger et de Klein-Gordon non linéaires. C. R. Acad. Sci. Paris Sér. I Math. 293, 489-492 (1981) Zbl 0492.35010 MR 0646873
3. [B] Bourgain, J.: Global Solutions of Nonlinear Schrödinger Equations. Amer. Math. Soc. Colloq. Publ. 46, Amer. Math. Soc. (1999) Zbl 0933.35178 MR 1691575
4. [BW] Bourgain, J.,Wang,W.: Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity. Ann. Scuola Norm. Sup. Pisa (4) 25, 197-215 (1997) Zbl 1043.35137 MR 1655515
5. [BusPer] Buslaev, V. S., Perelman, G. S.: Scattering for the nonlinear Schrödinger equation: states that are close to a soliton. Algebra i Analiz 4, no.6, 63-102 (1992) (in Russian); English transl.: St. Petersburg Math. J. 4, 1111-1142 (1993) Zbl 0853.35112 MR 1199635
6. [CazLio] Cazenave, T., Lions, P.-L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Comm. Math. Phys. 85, 549-561 (1982) Zbl 0513.35007 MR 0677997
7. [ChriKis] Christ, M., Kiselev, A.: Maximal functions associated with filtrations. J. Funct. Anal. 179, 409-425 (2001) Zbl 0974.47025 MR 1809116
8. [Cuc] Cuccagna, S.: Stabilization of solutions to nonlinear Schrödinger equations. Comm. Pure Appl. Math. 54, no.9, 1110-1145 (2001) Zbl 1031.35129 MR 2094268
9. [Flu] Flügge, S.: Practical Quantum Mechanics. Reprinting in one volume of Vols. I, II. Springer, New York (1974) Zbl 0934.81001 MR 0366248
10. [Froe] Fröhlich, J., Gustafson, S., Jonsson, B. L. G., Sigal, I. M.: Solitary wave dynamics in an external potential. Comm. Math. Phys. 250, 613-642 (2004) Zbl 1075.35075 MR 2094474
11. [GolSch] Goldberg, M., Schlag, W.: Dispersive estimates for Schrödinger operators in dimensions one and three. Comm. Math. Phys. 251, 157-178 (2004) Zbl 1086.81077 MR 2096737
12. [Gri] Grillakis, M.: Analysis of the linearization around a critical point of an infinite dimensional Hamiltonian system. Comm. Pure Appl. Math. 43, 299-333 (1990) Zbl 0731.35010 MR 1040143
13. [Har] Hartman, P.: Ordinary Differential Equations. Corrected reprint of the second (1982) edition. Classics in Appl. Math. 38, SIAM, Philadelphia, PA (2002) Zbl 1009.34001 MR 1929104
14. [Kat] Kato, T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258-279 (1965/1966) Zbl 0139.31203 MR 0190801
15. [KriSch1] Krieger, J., Schlag, W.: Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension. J. Amer. Math. Soc. 19, 815-920 (2006) Zbl pre05034395 MR 2219305
16. [KriSch2] Krieger, J., Schlag, W.: On the focusing critical semi-linear wave equation. Amer. J. Math. 129, 843-913 (2007) Zbl pre05170329 MR 2325106
17. [LiWig] Li, C., Wiggins, S.: Invariant Manifolds and Fibrations for Perturbed Nonlinear Schrödinger Equations. Appl. Math. Sci. 128, Springer, New York (1997) Zbl 0897.35070 MR 1475929
18. 2 critical nonlinear Schrödinger equation. Invent. Math. 156, 565-672 (2004) Zbl 1067.35110 MR 2061329
19. [Mur] Murata, M.: Asymptotic expansions in time for solutions of Schrödinger-type equations. J. Funct. Anal. 49, 10-56 (1982) Zbl 0499.35019 MR 0680855
20. [Per] Perelman, G.: On the formation of singularities in solutions of the critical nonlinear Schrödinger equation. Ann. Henri Poincaré 2, 605-673 (2001) Zbl 1007.35087 MR 1852922 (Pubitemid 33570417)
21. [Ra] Raphael, P.: Stability of the log-log bound for blow up solutions to the critical nonlinear Schrödinger equation. Math. Ann. 331, 577-609 (2005) Zbl 1082.35143 MR 2122541 (Pubitemid 40251235)
22. [ReeSim4] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Academic Press, New York (1979) Zbl 0401.47001 MR 0493421
23. [RodSch] Rodnianski, I., Schlag, W.: Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent. Math. 155, 451-513 (2004) Zbl 1063.35035 MR 2038194
24. [Sch1] Schlag, W.: Stable manifolds for an orbitally unstable NLS. Preprint (2004)
25. [Sch2] Schlag,W.: Dispersive estimates for Schrödinger operators: A survey. In: Mathematical Aspects of Nonlinear Dispersive Equations, Ann. of Math. Stud. 163, Princeton Univ. Press, 255-285 (2007) MR 2333215
26. [SogSmi] Smith, H., Sogge, C.: Global Strichartz estimates for nontrapping perturbations of the Laplacian. Comm. Partial Differential Equations 25, 2171-2183 (2000) Zbl 0972.35014 MR 1789924
27. [SofWei1] Soffer, A., Weinstein, M.: Multichannel nonlinear scattering for nonintegrable equations. Comm. Math. Phys. 133, 119-146 (1990) Zbl 0721.35082 MR 1071238
28. [SofWei2] Soffer, A.,Weinstein, M.: Multichannel nonlinear scattering, II. The case of anisotropic potentials and data. J. Differential Equations 98, 376-390 (1992) Zbl 0795.35073 MR 1170476
29. [St] Stein, E.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Univ. Press, Princeton, NJ (1993) Zbl 0821.42001 MR 1232192
30. [TsaYau] Tsai, T.-P., Yau, H.-T.: Stable directions for excited states of nonlinear Schrödinger equations. Comm. Partial Differential Equations 27, 2363-2402 (2002) Zbl 1021.35113 MR 1944033
31. [Wei1] Weinstein, M. I.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16, 472-491 (1985) Zbl 0583.35028 MR 0783974
32. [Wei2] Weinstein, M. I.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Comm. Pure Appl. Math. 39, 51-67 (1986) Zbl 0594.35005 MR 0820338