Articles

SKYRMIONS IN GROSS-PITAEVSKII FUNCTIONALS

  • Fanghua Lin ,
  • Taichia Lin ,
  • Juncheng Wei
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  • Courant Institute of Mathematical Sciences, New York University, New York, USA;Department of Mathematics, National Taiwan University, Taipei, Taiwan 106|Taida Institute of Mathematical Sciences (TIMS), Taipei, Taiwan, China;Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China

Received date: 2008-12-26

  Online published: 2009-05-20

Supported by

The research of FHL is partially supported by the NSF grant under DMS 0700517, TCL is partially supported by a research Grant from NSC and NCTS (National Center of Theoretical Sciences) of Taiwan, and JCW is partially supported by a General Research Fund from RGC of Hong Kong.

Abstract

In Bose-Einstein condensates (BECs), skyrmions can be characterized by pairs of linking vortex rings coming from two-component wave functions. Here we construct skyrmions by studying critical points of Gross-Pitaevskii functionals with two-component wave functions. Using localized energy method, we rigorously prove the existence, and describe the configurations of skyrmions in such BECs.

Cite this article

Fanghua Lin , Taichia Lin , Juncheng Wei . SKYRMIONS IN GROSS-PITAEVSKII FUNCTIONALS[J]. Acta mathematica scientia, Series B, 2009 , 29(3) : 751 -776 . DOI: 10.1016/S0252-9602(09)60069-2

References

[1] Anderson B P, Haljan P C, Regal C A, Feder D L, Collins L A, Clark C W, Cornell E A. Watching dark
solitons decay into vortex rings in a Bose-Einstein condensate. Phys Rev Lett, 2001, 86: 2926

[2] Battye R A, Cooper N R, Sutcliffe P M. Stable skyrmions in two-component Bose-Einstein condensates.
Phys Rev Lett, 2002, 88: 080401(1-4)

[3] Babaev E, Faddeev L D, Niemi A J. Hidden symmetry and knot solitons in a charged two-condensate Bose
system. Phys Rev B, 2002, 65: 100512(1-4)

[4] Chen X, Elliott C M, Qi T. Shooting method for vortex solutions of a complex-valued Ginzburg-Landau
equation. Proc Roy Soc Edinburgh Sect A, 1994, 124: 1075–1088

[5] Esteban M. A direct variational approach to Skyrme’s model for meson fields. Commun Math Phys, 1986,
105: 571–591; Erratum to: “A direct variational approach to Skyrme’s model for meson fields”. Comm
Math Phys, 2004, 251(1): 209–210

[6] Hervé R M, Hervé M. Étude qualitative des solutions réelles dúne équation diff´erentielle liée à l‘équation
de Ginzburg-Landau. Ann Inst H Poincaré Anal Non Linéaire, 1994, 11: 427–440


[7] Khawaja U A, Stoof H. Skyrmions in a ferromagnetic Bose-Einstein condensate. Nature, 2001, 411:
918–920

[8] Leggett A J. Bose-Einstein condensation in the alkali gases: Some fundamental concepts. Rev Mod Phys,
2001, 73: 307–356

[9] Lin F, Yang Y. Existence of two-dimensional skyrmions via the concentration-compactness method. Comm
Pure Appl Math, 2004, 57(10): 1332–1351

[10] Lin Fanghua, Yang Yisong. Energy splitting, substantial inequality, and minimization for the Faddeev and
Skyrme models. Comm Math Phys, 2007, 269(1): 137–152

[11] Lin T C. The stability of the radial solution to the Ginzburg-Landau equation. Communication in Partial
Differential Equations, 1997, 22(3/4): 619–632

[12] Frank Pacard, Tristan Riviere. Linear and nonlinear aspects of vortices. The Ginzburg-Landau model.
Progress in Nonlinear Differential Equations and their Applications, 39. Boston, MA: Birkhauser Boston,
Inc, 2000

[13] Pitaevskii L, Stringari S. Bose-Einstein Condensation. Oxford: Oxford University Press, 2003

[14] Rajaraman R. Solitons and Instantons. Amsterdam: North-Holland, 1989

[15] Ruostekoski J, Anglin J R. Creating vortex rings and three-dimensional skyrmions in Bose-Einstein con-
densates. Phys Rev Lett, 2001, 86: 3934(1-4)

[16] Savage C M, Ruostekoski J. Energetically stable particlelike skyrmions in a trapped Bose-Einstein con-
densate. Phys Rev Lett, 2003, 91: 010403(1-4)

[17] Skyrme T H R. A nonlinear-field theory. Proc R Soc London A, 1961, 260: 127–138

[18] Timmermans E. Phase separation of Bose-Einstein condensates. Phys Rev Lett, 1998, 81: 5718–5721

[19] Wei J. On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet prob-
lem. J Diff Eqns, 1996, 129: 315–333

[20] Wei J. On interior spike layer solutions for some sin gilar perturbation problems. Proc Royal Soc Edinburgh
Section A, 1998, 128: 849–974

[21] Wei J. Existence and stability of spikes for the Gierer-Meinhardt system//Chipot M, ed. Handbook of
Differential Equations: Stationary Partial Differential Equations, Volume 5. Elsevier, 2008: 487–585

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