Scattering theory for the grosspitaevskii equation in three dimensions

Communications in Contemporary Mathematics

Volumn 11, Issue 4, 2009, Pages 657-707

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

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

^b KYOTO UNIVERSITY   (Japan)

Author keywords

GrossPitaevskii equation; Nonlinear Schr dinger equation; Scattering theory

Indexed keywords

EID: 69849094134     PISSN: 02191997     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0219199709003491     Document Type: Article

References (38)

1. J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Grundlehren Math. Wiss., Vol.223 (Springer, 1976).
2. F. Bethuel, P. Gravejat and J. C. Saut, Travelling waves for the Gross-Pitaevskii equation II, Comm. Math. Phys. 285(2) (2009) 567-651.
3. F. Bethuel, G. Orlandi and D. Smets, Vortex rings for the Gross-Pitaevskii equation, J. Eur. Math. Soc. 6(1) (2004) 17-94.
4. F. Bethuel and J. C. Saut, Travelling waves for the Gross-Pitaevskii equation. I, Ann. Inst. H. Poincaré 70(2) (1999) 147-238.
5. F. Bethuel and J. C. Saut, Vortices and sound waves for the Gross-Pitaevskii equation, in Nonlinear PDE's in Condensed Matter and Reactive Flows, NATO Sci. Ser. C Math. Phys. Sci., Vol.569 (Kluwer Acad. Publ., Dordrecht, 2002), pp. 339-354.
6. F. Bethuel and D. Smets, A remark on the Cauchy problem for the 2D Gross- Pitaevskii equation with nonzero degree at infinity, Differ. Integral Equ. 20(3) (2007) 325-338.
7. D. Chiron, Travelling waves for the Gross-Pitaevskii equation in dimension larger than two, Nonlinear Anal. 58(1-2) (2004) 175-204.
8. R. Coifman and Y. Meyer, Commutateurs díntegrales singulieres et operateurs multilineaires, Ann. Inst. Fourier (Grenoble) 28(3) (1978) xi, 177-202.
9. J. E. Colliander and R. L. Jerrard, Vortex dynamics for the Ginzburg-Landau- Schr̈odinger equation, Int. Math. Res. Not. 1998(7) (1988) 333-358.
10. J. E. Colliander and R. L. Jerrard, Ginzburg-Landau vortices: Weak stability and Schr̈odinger equation dynamics, J. Anal. Math. 77 (1999) 129-205.
11. A. L. Fetter and A. A. Svidzinsky, Vortices in a trapped dilute Bose-Einstein condensate, J. Phys. Condens. Matter 13 (2001) R135-R194.
12. C. Gallo, Schr̈odinger group on Zhidkov spaces, Adv. Differential Equations 9(5-6) (2004) 509-538.
13. P. Gérard, The Cauchy problem for the Gross-Pitaevskii equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 23(5) (2006) 765-779.
14. P. Gérard, The Gross-Pitaevskii equation in the energy space, to appear in Thematics Program on Gross-Pitaevskii Equation, Contemporary Mathematics (Amer. Math. Soc.).
15. P. Germain, N. Masmoudi and J. Shatah, Global solutions for 3D quadratic Schr̈odinger equations, Int. Math. Res. Not. 2009(3) (2009) 414-432.
16. J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schr̈odinger equations, J. Math. Pures Appl. (9) 64(4) (1985) 363-401.
17. L. Grafakos and N. J. Kalton, The Marcinkiewicz multiplier condition for bilinear operators, Studia Math. 146(2) (2001) 115-156.
18. P. Gravejat, A non-existence result for supersonic travelling waves in the Gross- Pitaevskii equation, Comm. Math. Phys. 243(1) (2003) 93-103.
19. P. Gravejat, Asymptotics for the travelling waves in the Gross-Pitaevskii equation, Asymptot. Anal. 45 (2005) 227-299.
20. S. Gustafson, K. Nakanishi and T.-P. Tsai, Scattering theory for the Gross-Pitaevskii equation, Math. Res. Lett. 13(2) (2006) 273-285.
21. S. Gustafson, K. Nakanishi and T.-P. Tsai, Global dispersive solutions for the Gross- Pitaevskii equation in two and three dimensions, Ann. Henri Poincaré 8 (2007) 1303-1331.
22. N. Hayashi, T. Mizumachi and P. I. Naumkin, Time decay of small solutions to quadratic nonlinear Schr̈odinger equations in 3D, Differ. Integral Equations 16(2) (2003) 159-179.
23. N. Hayashi and P. I. Naumkin, On the quadratic nonlinear Schr̈odinger equation in three space dimensions, Int. Math. Res. Not. 2000(3) (2000) 115-132.
24. N. Hayashi and P. I. Naumkin, Asymptotics in time of solutions to nonlinear Schr̈odinger equations in two space dimensions, Funkcial. Ekvac. 49(3) (2006) 415-425.
25. R. L. Jerrard, Vortex filament dynamics for Gross-Pitaevsky type equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. 1(4) (2002) 733-768.
26. F. John, Blow-up for quasilinear wave equations in three space dimensions, Comm. Pure Appl. Math. 34(1) (1981) 29-51.
27. C. A. Jones and P. H. Roberts, Motions in a Bose condensate: IV. Axisymmetric solitary waves, J. Phys. A 15 (1982) 2599-2619.
28. C. A. Jones, S. J. Putterman and P. H. Roberts, Motions in a Bose condensate: V. Stability of solitary wave solutions of non-linear Schr̈odinger equations in two and three dimensions, J. Phys. A 19 (1986) 2991-3011.
29. O. Lange and B. J. Schroers, Unstable manifolds and Schr̈odinger dynamics of Ginzburg-Landau vortices, Nonlinearity 15(5) (2002) 1471-1488.
30. F. H. Lin and J. X. Xin, On the incompressible fluid limit and the vortex motion law of the nonlinear Schr̈odinger equation, Comm. Math. Phys. 200(2) (1999) 249-274.
31. J. E. Lin and W. A. Strauss, Decay and scattering of solutions of a nonlinear Schr̈odinger equation, J. Funct. Anal. 30(2) (1978) 245-263.
32. K. Nakanishi, Asymptotically-free solutions for the short-range nonlinear Schr̈odinger equation, SIAM J. Math. Anal. 32(6) (2001) 1265-1271.
33. Y. N. Ovchinnikov and I. M. Sigal, Long-time behaviour of Ginzburg-Landau vortices, Nonlinearity 11(5) (1998) 1295-1309.
34. J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math. 38(5) (1985) 685-696.
35. A. Shimomura and Y. Tsutsumi, Nonexistence of scattering states for some quadratic nonlinear Schrodinger equations in two space dimensions, Differ. Integral Equations 19(9) (2006) 1047-1060.
36. D. Spirn, Vortex motion law for the Schr̈odinger-Ginzburg-Landau equations, SIAM J. Math. Anal. 34(6) (2003) 1435-1476.
37. C. Sulem and P.-L. Sulem, The Nonlinear Schr̈odinger Equation. Self-Focusing and Wave Collapse, AppliedMathematical Sciences, Vol.139 (Springer-Verlag, New York, 1999).
38. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, 2nd edn. (Johann Ambrosius Barth, Heidelberg, 1995).