Scattering for the Gross-pitaevskii equation

Mathematical Research Letters

Volumn 13, Issue 2-3, 2006, Pages 273-285

  Gustafson, Stephen ^a   Nakanishi, Kenji ^b,c   Tsai, Tai Peng ^a  

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

^b NAGOYA UNIVERSITY   (Japan)

^c KYOTO UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33746109034     PISSN: 10732780     EISSN: None     Source Type: Journal    
DOI: 10.4310/MRL.2006.v13.n2.a8     Document Type: Article

References (30)

1. J. Bergh and J. Löfström, Interpolation spaces, An introduction. Grundlehren Math. Wiss. Vol. 223, Springer, Berlin-Heiderberg-New York, 1976.
2. F. Bethuel, G. Orlandi, and D. Smets, Vortex rings for the Gross-Pitaevakii equation. J. Euro. Math. Soc. 6 (2004), no. 1, 17-94.
3. F. Bethuel and J. C. Saut, Travelling waves for the Gross-Pitaevskii equation. I. Ann. Inst. H. Poincaré Phys. Théor, 70 (1999), no. 2, 147-238.
4. D. Chiron, Travelling waves for the Gross-Pitaevskii equation in dimension larger than two. Nonlinear Anal. 58 (2004), no. 1-2, 175-204.
5. 8. Nonlinear Anal. 14 (1990), no. 10, 807-836.
6. J. E. Colliander and R. L. Jerrard, Vortex dynamics for the Ginzburg-Landau-Schrödinger equation. Internat. Math. Res. Notices 1908, no. 7, 333-358;
7. and Ginzburg-Landau vortices: weak stability and Schrödinger equation dynamics. J. Anal. Math. 77 (1999), 129-205.
8. A.L. Fetter and A.A. Svidzinsky, Vortices in a trapped dilute Bose-Einstein condensate. Preprint: arXiv:cond-mat/0102003.
9. J. Ginibre and T. Ozawa, Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension n ≥ 2. Comm. Math. Phys. 151 (1993), no. 3, 619-645.
10. J. Ginibre, T. Ozawa, and G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor. 60 (1994), no. 2, 211-239.
11. J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations. J. Math. Pures Appl. (9) 64 (1985), no. 4, 363-401.
12. P. Gravejat, Limit at infinity for travelling waves in the Gross-Pitaevskii equation. C. R. Math. Acad. Sci. Paris 336 (2003), no. 2, 147-152;
13. and A non-existence result for supersonic travelling waves in the Gross-Pitaevskii equation. Comm. Math. Phys. 243 (2003), no. 1, 93-103.
14. S. Gustafson, Dynamic stability of magnetic vortices. Nonlinearity 15 (2002), no. 5, 1717-1728.
15. S. Gustafson, I.M. Sigal, Effective dynamics of magnetic vortices. To appear in Adv. Math. (2005).
16. N. Hayashi, T. Mizumachi and P. I. Naumkin, Time decay of small solutions to quadratic nonlinear Schrödinger equations in 3D. Differential Integral Equations 16 (2003), no. 2, 159-179.
17. N. Hayashi, P. I. Naumkin, A. Shimomura and S. Tonegawa, Modified wave operators for nonlinear Schrödinger equations in one and two dimensions. Electron. J. Differential Equations 2004, No. 62, 16 pp.
18. R. Jerrard, Vortex dynamics for the Ginzburg-Landau wave equation. Calc. Var. Partial Diff. Eqns. 9 (1999), no.8, 683-688.
19. _, Vortex filament dynamics for Gross-Pitaevsky type equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. 1 (2002), no. 4, 733-768.
20. C.A. Jones, S.J. Putterman, P.H. Roberts, Motions in a Bose condensate: stability of solitary wave solutions of non-linear Schrödinger equations in two and three dimensions. J. Phys. A: Math. Gen. 19 (1986), 2991-3011.
21. M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no.5, 955-980.
22. O. Lange and B. J. Schroers, Unstable manifolds and Schrödinger dynamics of Ginzburg-Landau vortices, Nonlinearity 15 (2002), no. 5, 1471-1488.
23. F.-H. Lin, Vortex dynamics for the nonlinear wave equation, Comm. Pure Appl. Math. 52 (1999), no.6, 737-429.
24. F. H. Lin and J. X. Xin, On the incompressible fluid limit and the vortex motion law of the nonlinear Schrödinger equation, Comm. Math. Phys. 200 (1999), no. 2, 249-274.
25. Y. N. Ovchinnikov and I. M. Sigal, Long-time behaviour of Ginzburg-Landau vortices. Nonlinearity 11 (1998), no. 5, 1295-1309.
26. T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension, Comm. Math. Phys. 139 (1991), no. 3, 479-493.
27. 1+4, preprint, http://jp.arxiv.org/abs/math.AP/0501462.
28. A. Shimomura, Nonexistence of asymptotically free solutions for quadratic nonlinear Schrödinger equations in two space dimensions. Differential Integral Equations 18 (2005), no. 3, 325-335.
29. D. Spirn, Vortex motion law for the Schrödinger-Ginzburg-Landau equations, SIAM J. Math. Anal. 34 (2003), no. 6, 1435-1476.
30. Y. Tsutsumi, Stability of constant equilibrium for the Maxwell-Higgs equations, Funkcial. Ekvac. 46 (2003), no. 1, 41-62.