Vortex rings for the Gross-Pitaevskii equation

Journal of the European Mathematical Society

Volumn 6, Issue 1, 2004, Pages 17-94

  Bethuel, F ^a   Orlandi, G ^b   Smets, D ^a  

^a UNIVERSITÉ PIERRE ET MARIE CURIE   (France)

^b UNIVERSITY OF VERONA   (Italy)

Author keywords

Ginzburg Landau; Gross Pitaevskii; NLS; Vortex rings

Indexed keywords

EID: 2442530451     PISSN: 14359855     EISSN: None     Source Type: Journal    
DOI: 10.4171/JEMS/2     Document Type: Article

References (42)

1. Alberti, G., Baldo, S., Orlandi, G.: Variational convergence for functionals of Ginzburg-Landau type. Preprint (2002)
2. Allard, W.: On the first variation of a varifold. Ann. of Math. 95, 417-491 (1972) Zbl 0252.49028 MR 46 #6136
3. Almgren, F.: Optimal isoperimetric inequalities. Indiana Univ. Math. J. 35, 451-547 (1986) Zbl 0585.49030 MR 88c:49032
4. Ambrosetti, A., Struwe, M.: Existence of steady vortex rings in an ideal fluid. Arch. Rat. Mech. Anal. 108, 97-109 (1989) Zbl 0694.76012 MR 90g:76036
5. Ambrosio, L., Soner, M.: A measure theoretic approach to higher codimension mean curvature flow. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25, 27-49 (1997) Zbl pre01766596 MR 99m:35217
6. Benjamin, T. B.: The stability of solitary waves. Proc. Roy. Soc. London Ser. A 328, 153-183 (1972) MR 49 #3348
7. 1/2. C. R. Acad. Sci. Paris Sér. I Math. 333, 1-8 (2001) Zbl pre01719347
8. Bethuel, F., Brezis, H., Hélein, F.: Asymptotics for the minimizers of a Ginzburg-Landau functional. Calc. Var. PDE 1, 123-148 (1993) Zbl 0834.35014
9. Bethuel, F., Brezis, H., Hélein, F.: Ginzburg-Landau Vortices. Birkhäuser, Boston (1994) Zbl 0802.35142 MR95c:58044
10. Bethuel, F., Brezis, H., Orlandi, G.: Asymptotics for the Ginzburg-Landau equation in arbitrary dimensions. J. Fund. Anal. 186, 432-520 (2001). Erratum 188, 548-549 (2002) Zbl pre01703083 MR 2002k:35081a MR 2002k:35081b
11. Bethuel, F., Orlandi, G.: Uniform estimates for the parabolic Ginzburg-Landau equation. ESAIM Control Optim. Calc. Var. 8, 219-238 (2002) MR 1 932 951
12. Bethuel, F., Orlandi, G., Smets, D.: Convergence of the parabolic Ginzburg-Landau equation to motion by mean curvature. C. R. Acad. Sci. Paris Sér. I Math. 336, 217-222 (2003), and preprint R03004 Lab. J. L. Lions (2003).
13. Bethuel, F., Rivière, T.: A minimization problem related to superconductivity. Ann. Inst. H. Poincaré Anal. Non Linéaire 12, 243-303 (1995) Zbl 0842.35119 MR 96g:35045
14. Bethuel, F., Saut, J.-C.: Traveling waves for the Gross-Pitaevskii equation. Ann. Inst. H. Poincaré Phys. Théor. 70, 147-238 (1999) MR 2000d:35211
15. Bona, J.: On the stability theory of solitary waves. Proc. Roy. Soc. London Ser. A 344, 363-374 (1975) Zbl 0328.76016 MR 52 #7292
16. 1/2 with values into the circle. C. R. Acad. Sci. Paris Sér. I Math. 331, 119-124 (2000), and detailed paper in preparation.
17. Brezis, H., Mironescu, P.: Sur une conjecture de E. De Giorgi relative à l'énergie de Ginzburg-Landau. C. R. Acad. Sci. Paris Sér. I Math. 319, 167-170 (1994) Zbl 0805.49004 MR 95h:49016
18. Cazenave, T., Lions, P.-L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Comm. Math. Phys. 85, 549-561 (1982) Zbl 0513.35007 MR 841:81015
19. Farina, A.: From Ginzburg-Landau to Gross-Pitaevskii. Preprint (2002)
20. Federer, H.: Geometric Measure Theory. Springer, Berlin (1969) Zbl 0176.00801 MR 41 #1976
21. Fetter, A. L., Svidzinsky, A. A.: Vortices in a trapped dilute Bose-Einstein condensate.J. Phys.: Condens. Matter 13, R135-R194 (2001)
22. Fraenkel, L. E., Berger, M. S.: A global theory of steady vortex rings in an ideal fluid. Acta Math. 132, 13-51 (1974) Zbl 0282.76014 MR 54 #10901
23. Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton Univ. Press (1983) Zbl 0516.49003 MR 86b:49003
24. Ginibre, J., Velo, G.: The Cauchy problem in local spaces for the complex Ginzburg-Landau equation II. Contraction methods. Comm. Math. Phys. 187, 45-79 (1997) Zbl 0889.35046 MR 981:35174
25. Gravejat, P.: Decay for travelling waves in the Gross-Pitaevskii equation. Preprint
26. Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry. I and II. J. Funct. Anal. 74, 160-197 (1987) and 94, 308-348 (1990) Zbl 0656.35122 Zbl 0711.58013 MR 88g:35169 MR 92a:35135
27. Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry. I and II. J. Funct. Anal. 74, 160-197 (1987) and 94, 308-348 (1990) Zbl 0656.35122 Zbl 0711.58013 MR 88g:35169 MR 92a:35135
28. Helmholtz, H.: Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelwegungen entsprechen. J. Reine Angew. Math. 55, 25-55 (1858)
29. Jerrard, R. L.: Vortex filament dynamics for Gross-Pitaevsky like equations, I. Preprint
30. Jerrard, R. L., Soner, H. M.: The Jacobian and the Ginzburg-Landau energy. Calc. Var. PDE 14, 151-191 (2002) Zbl pre01 756808 MR 2003d:35069
31. Jones, C. A., Roberts, P. H.: Motions in a Bose condensate IV. Axisymmetric solitary waves, J. Phys. A 15, 2599-2619 (1982)
32. Lin, F. H., Rivière, T.: Complex Ginzburg-Landau equation in high dimensions and codimension two area minimizing currents. J. Eur. Math. Soc. 1, 237-311 (1999). Erratum, 2, 87-91 (2000) Zbl 0939.35056 MR 2000g:49048
33. Lin, F. H., Rivière, T.: A quantization property for static Ginzburg-Landau vortices. Comm. Pure Appl. Math. 54, 206-228 (2001) Zbl pre01592012 MR 2001k:35097
34. Moore, D. W., Pullin, D. I.: On steady compressible flows with compact vorticity; the compressible Hill's spherical vortex. J. Fluid Mech. 374, 285-303 (1998) Zbl 0936.76075 MR 991:76136
35. Morrey, C. B.: Multiple Integrals in the Calculus of Variations. Springer, New York (1966) Zbl 0142.38701 MR 34 #2380
36. Newell, A. C., Moloney, J. V.: Nonlinear Optics. Addison-Wesley (1992)
37. Nozières, P., Pines, D.: The Theory of Quantum Liquids, Vol. II: Superfluid Bose Liquids. Perseus Books, Reading, MA (1999)
38. Rivière, T.: Line vortices in the U (1) Higgs model. ESAIM Control Optim. Calc. Var. 1, 77-167 (1996) Zbl 0874.53019 MR 97g:58043
39. Roberts, P. H., Berloff, N. G.: The nonlinear Schrödinger equation as a model for superfluidity. Preprint
40. Simon, L.: Lectures on Geometric Measure Theory. Proc. Centre for Math. Anal., Austr. Nat. Univ. (1983) Zbl 0546.49019 MR 87a:49001
41. Struwe, M.: On the asymptotic behavior of the Ginzburg-Landau model in 2 dimensions. J. Diff. Equ. 7, 1613-1624 (1994), Erratum 8, 224 (1995) Zbl 0809.35031 MR 95g:35057a
42. Sulem, C., Sulem, P.-L.: The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Appl. Math. Sci. 139, Springer (1999) Zbl 0928.35157 MR 2000f:35139