A centre-stable manifold for the focussing cubic NLS in ℝ1+3

Communications in Mathematical Physics

Volumn 280, Issue 1, 2008, Pages 145-205

  Beceanu, Marius ^a  

^a UNIVERSITY OF CHICAGO   (United States)

Author keywords

[No Author keywords available]

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EID: 43349101598     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-008-0427-3     Document Type: Article

References (48)

1. Agmon S. (1975). Spectral properties of Schrdinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2(2): 151-218
2. Bates P.W. and Jones C.K.R.T. (1989). Invariant manifolds for semilinear partial differential equations. Dynamics Reported 2: 1-38
3. Berestycki H. and Cazenave T. (1981). Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires. C. R. Acad. Sci. Paris Sér. I Math. 293(9): 489-492
4. Berestycki H. and Lions P.L. (1983). Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rat. Mech. Anal. 82(4): 313-345
5. Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Berlin-Heidelberg-New York: Springer-Verlag, 1976
6. Buslaev, V.S., Perelman, G.S.: Scattering for the nonlinear Schrödinger equation: states that are close to a soliton (Russian). Algebra i Analiz 4(6), 63-102 (1992);
7. translation in St. Petersburg Math. J. 4(6), 1111-1142 (1993)
8. Buslaev, V.S., Perelman, G.S.: On the stability of solitary waves for nonlinear Schrödinger equations. In: Nonlinear evolution equations, Amer. Math. Soc. Transl. Ser. 2, 164, Providence, RI: Amer. Math. Soc. 1995, pp. 75-98
9. Buslaev, V.S., Perelman, G.S.: Nonlinear scattering: states that are close to a soliton (Russian). Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov, (POMI) 200 (1992)
10. Kraev. Zadachi Mat. Fiz. Smezh. Voprosy Teor. Funktsii, 24, 38-50, 70, 187;
11. translation in J. Math. Sci. 77(3), 3161-3169 (1995)
12. Cazenave, T.: Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, Providence, RI: Amer. Math. Soc., 2003
13. Cazenave T. and Lions P.L. (1982). Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85: 549-561
14. 3 = 0 and a variational characterization of other solutions. Arch. Rat. Mech. Anal. 46: 81-95
15. Cuccagna S. (2001). Stabilization of solutions to nonlinear Schrödinger equations. Comm. Pure Appl. Math. 54(9): 1110-1145
16. Cuccagna S., Pelinovsky D. and Vougalter V. (2005). Spectra of positive and negative energies in the linearized NLS problem. Comm. Pure Appl. Math. 58(1): 1-29
17. Demanet L. and Schlag W. (2006). Numerical verification of a gap condition for a linearized NLS equation. Nonlinearity 19: 829-852
18. Erdogan, B., Schlag, W.: Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II. To appear in Journal d'Analyse Mathematique, available at http://arxiv.org/list/math/0504585, 2005
19. Gesztesy F., Jones C.K.R.T., Latushkin Y. and Stanislavova M. (2000). A spectral mapping theorem and invariant manifolds for nonlinear Schrödinger equations. Indiana Univ. Math. J. 49(1): 221-243
20. Glassey R.T. (1977). On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation. J. Math. Phys. 18: 1794-1797
21. Grillakis M., Shatah J. and Strauss W. (1987). Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal. 74(1): 160-197
22. Grillakis M., Shatah J. and Strauss W. (1990). Stability theory of solitary waves in the presence of symmetry. II. J. Funct. Anal. 94(1): 308-348
23. Hundertmark D. and Lee Y.-R. (2007). Exponential decay of eigenfunctions and generalized eigenfunctions of a non self-adjoint matrix Schrödinger operator related to NLS. Bull. London Math. Soc. 39(5): 709-720
24. Keel M. and Tao T. (1998). Endpoint Strichartz estimates. Amer. Math. J. 120: 955-980
25. Kenig C. and Merle F. (2006). Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case. Inv. Math. 166(3): 645-675
26. Kelley P.L. (1965). Self-focusing of optical beams. Phys. Rev. Lett. 15: 1005-1008
27. Krieger J. and Schlag W. (2006). Stable manifolds for all monic supercritical NLS in one dimension. J. AMS 19(4): 815-920
28. Krieger, J., Schlag, W.: Non-generic blow-up solutions for the critical focusing NLS in 1-d. http://arxiv.org/list/math/0508576, 2005
29. Krieger J. and Schlag W. (2007). On the focusing critical semi-linear wave equation. Amer. J. Math. 129(3): 843-913
30. n) Arch. Rat. Mech. Anal. 65: 243-266
31. n) Arch. Rat. Mech. Anal. 99: 115-145
32. Merle F. (1993). Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power. Duke Math. J. 69(2): 427-454
33. 2 critical nonlinear Schrödinger equation. J. Amer. Math. Soc. 19(1): 37-90
34. Perelman G. (2001). On the formation of singularities in solutions of the critical nonlinear Schrödinger equation. Ann. Henri Poincaré 2(4): 605-673
35. Pillet C.A. and Wayne C.E. (1997). Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations. J. Diff. Eq. 141(2): 310-326
36. Rodnianski I. and Schlag W. (2004). Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent. Math. 155(3): 451-513
37. Rodnianski I., Schlag W. and Soffer A. (2004). Dispersive analysis of charge transfer models. Comm. Pure and Appl. Math. 58(2): 149-216
38. Rodnianski, I., Schlag, W., Soffer, A.: Asymptotic stability of N-soliton states of NLS. http://arxiv.org/list/math/0309114, 2003
39. Schlag, W.: Stable Manifolds for an orbitally unstable NLS. http://arxiv.org/list/math/0405435, 2004
40. Schlag W. (2006). Spectral theory and nonlinear partial differential equations: a survey. Discrete Contin. Dyn. Syst. 15(3): 703-723
41. Soffer A. and Weinstein M.I. (1990). Multichannel nonlinear scattering for nonintegrable equations. Comm. Math. Phys. 133: 119-146
42. Soffer A. and Weinstein M.I. (1992). Multichannel nonlinear scattering, II. The case of anisotropic potentials and data. J. Diff. Eq. 98: 376-390
43. Stein E. (1994). Harmonic Analysis. Princeton University Press, Princeton, NJ
44. Sulem, C., Sulem, P.-L.: The Nonlinear Schrödinger Equation. Self-focusing and Wave Collapse. Applied Mathematical Sciences, 139, New York: Springer-Verlag, 1999
45. Taylor, M.E.: Tools for PDE. Pseudodifferential operators, paradifferential operators, and layer potentials. Mathematical Surveys and Monographs, 81, Providence, RI: Amer. Math. Soc., 2000
46. Weinstein M.I. (1986). Lyapunov stability of ground states of nonlinear dispersive equations. Comm. Pure Appl. Math. 39(1): 51-67
47. Weinstein M.I. (1985). Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16(3): 472-491
48. Yajima K. (2005). Dispersive estimate for Schrödinger equations with threshold resonance and eigenvalue. Commun. Math. Phys. 259(2): 475-509