On the wave operators for the critical nonlinear Schrödinger equation

Mathematical Research Letters

Volumn 15, Issue 1, 2008, Pages 185-195

  Carles, Rémi ^a   Ozawa, Tohru ^b  

^a CNRS   (France)

^b HOKKAIDO UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 41149125148     PISSN: 10732780     EISSN: None     Source Type: Journal    
DOI: 10.4310/MRL.2008.v15.n1.a15     Document Type: Article

References (19)

1. 2-critical NLS, Commun. Pure Appl. Anal. 5 (2006), no. 4, 691-708.
2. R. Carles, Geometric optics with caustic crossing for some nonlinear Schrödinger equations, Indiana Univ. Math. J. 49 (2000), no. 2, 475-551.
3. _, Remarques sur les mesures de Wigner, C. R. Acad. Sci. Paris, t. 332, Série I 332 (2001), no. 11, 981-984.
4. _, Cascade of phase shifts for nonlinear Schrödinger equations, J. Hyperbolic Differ. Equ. 4 (2007), no. 2, 207-231.
5. T. Cazenave, Semilinear Schrödinger equations, Vol. 10 of Courant Lecture Notes in Mathematics, New York University Courant Institute of Mathematical Sciences, New York (2003).
6. T. Cazenave and F. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, in Lect. Notes in Math., Vol. 1394, 18-29, Springer-Verlag, Berlin (1989).
7. _, Rapidly Decaying Solutions of the Nonlinear Schrödinger Equation, Comm. Math. Phys. 147 (1992) 75-100.
8. P. Gérard, Oscillations and concentration effects in semilinear dispersive wave equations, J. Funct. Anal. 141 (1996), no. 1, 60-98.
9. J. Ginibre, An introduction to nonlinear Schrödinger equations, in R. Agemi, Y. Giga, and T. Ozawa, editors, Nonlinear waves (Sapporo, 1995), GAKUTO International Series, Math. Sciences and Appl., 85-133, Gakkōtosho, Tokyo (1997).
10. J. Ginibre, T. Ozawa, and G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. IHP (Physique Théorique) 60 (1994) 211-239.
11. J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. II Scattering theory, general case, J. Funct. Anal. 32 (1979) 33-71.
12. _, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl. (9) 64 (1985), no. 4, 363-401.
13. S. Katayama and Y. Tsutsumi, Global existence of solutions for nonlinear Schrödinger equations in one space dimension, Comm. Partial Differential Equations 19 (1994), no. 11-12, 1971-1997.
14. T. Ozawa, On the nonlinear Schrödinger equations of derivative type, Indiana Univ. Math. J. 45 (1996), no. 1, 137-163.
15. T. Ozawa and Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations, Differential Integral Equations 11 (1998), no. 2, 201-222.
16. M. Tanaka, Nonlinear self-modulation problem of the Benjamin-Ono equation, J. Phys. Soc. Japan 51 (1982), no. 8, 2686-2692.
17. T. Tao, M. Visan, and X. Zhang, Global well-posedness and scattering for the mass-critical nonlinear Schrödinger equation for radial data in high dimensions, Duke Math. J. 140 (2007), no. 1, 165-202.
18. Y. Tsutsumi, Scattering problem for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 43 (1985), no. 3, 321-347.
19. _, The null gauge condition and the one-dimensional nonlinear Schrödinger equation with cubic nonlinearity, Indiana Univ. Math. J. 43 (1994), no. 1, 241-254.