On scattering for NLS: From euclidean to hyperbolic space

Discrete and Continuous Dynamical Systems

Volumn 24, Issue 4, 2009, Pages 1113-1127

  Banica, Valeria ^a   Carles, Rémi ^b,c   Duyckaerts, Thomas ^d  

^a UNIVERSITÉ D EVRY VAL D ESSONNE   (France)

^b UNIV MONTPELLIER   (France)

^c CNRS   (France)

^d UNIVERSITÉ DE CERGY PONTOISE   (France)

Author keywords

Hyperbolic space; Nonlinear Schr dinger equation; Scattering theory

Indexed keywords

EID: 69549108626     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2009.24.1113     Document Type: Article

References (27)

1. J.-P. Anker and V. Pierfelice, Nonlinear Schrodinger equation on real hyperbolic spaces, preprint, http: //arxiv. org/pdf/0801.3523
2. H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.
3. H. Bahouri, P. Gérard, and C.-J. Xu, Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg, J. Anal. Math., 82 (2000), 93-118.
4. V. Banica, The nonlinear Schrödinger equation on the hyperbolic space, Comm. Partial Differential Equations, 32 (2007), 1643-1677.
5. V. Banica, R. Carles and G. Staffilani, Scattering theory for radial nonlinear Schrödinger equations on hyperbolic space, Geom. Punct. Anal., 18 (2008), 367-399.
6. V. Banica and T. Duyckaerts, Weighted Strichartz estimates for radial Schrodinger equation on noncompact manifolds, Dyn. Partial Differ. Equ., 4 (2007), 335-359.
7. J. E. Barab, Nonexistence of asymptotically free solutions for nonlinear Schrödinger equation, J. Math. Phys., 25 (1984), 3270-3273.
8. J.-F. Bony, R. Carles, D. Häfner and L. Michel, Scattering theory for the Schrödinger equation with repulsive potential, J. Math. Pures Appl., 84 (2005), 509-579.
9. N. Burq, F. Planchon, J. G. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003), 519-549. (Pubitemid 37195024)
10. R. Carles, Nonlinear Schrödinger equations with repulsive harmonic potential and applications, SIAM J. Math. Anal., 35 (2003), 823-843.
11. T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics, vol.10, New York University Courant Institute of Mathematical Sciences, New York, 2003.
12. J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on ℝ3, Comm. Pure Appl. Math., 57 (2004), 987-1014.
13. J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in ℝ3, Ann. of Math. (2), 167 (2008), 767-865.
14. J. Dereziński and C. Gérard, "Scattering Theory of Quantum And Classical N-particle Systems," Texts and Monographs in Physics, Springer Verlag, Berlin Heidelberg, 1997.
15. N. Dunford and J. T. Schwartz, "Linear operators. Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space," With the assistance of William G. Bade and Robert G. Battle, Interscience Publishers John Wiley & Sons New York-London, 1963.
16. J. Ginibre, An introduction to nonlinear Schrödinger equations, in "Nonlinear Waves (Sapporo, 1995)" (R. Agemi, Y. Giga and T. Ozawa, eds.), GAKUTO International Series, Math. Sciences and Appl., Gakkōtosho, Tokyo, 1997, 85-133.
17. J. Ginibre and G. Velo, On a class of nonlinear Schrodinger equations. I The Cauchy problem, general case, J. Funct. Anal., 32 (1979), 1-32.
18. J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrodinger equations, J. Math. Pures Appl. (9), 64 (1985), 363-401.
19. A. Hassell, T. Tao and J. Wunsch, A Strichartz inequality for the Schrödinger equation on nontrapping asymptotically conic manifolds, Comm. Partial Differential Equations, 30 (2005), 157-205.
20. A. Ionescu and G. Staffilani, Semilinear Schrödinger flows on hyperbolic spaces: Scattering in H1, preprint, http://arxiv.org/pdf/0801.2957
21. P. Petersen, "Riemannian Geometry," Graduate Texts in Math., vol.171, Springer-Verlag, New York, 1998. xvi+432 pp.
22. V. Pierfelice, Weighted Strichartz estimates for the Schrödinger and wave equations on Damek-Ricci spaces, Math. Z., 260 (2008), 377-392.
23. E. Ryckman and M. Visan, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R1+4, Amer. J. Math., 129 (2007), 1-60.
24. W. A. Strauss, Nonlinear scattering theory, in "Scattering theory in mathematical physics," Proc. NATO advanced Study Inst., Denver 1973, 53-78 (1974).
25. T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343.
26. M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., 138 (2007), 281-374.
27. D. Yafaev, "Scattering Theory: Some Old and New Problems," Lecture Notes in Mathematics, vol.1735, Springer-Verlag, Berlin, 2000.