Local minimizers with vortex filaments for a Gross-Pitaevsky functional

Jerrard, Robert L.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 1, pp. 35-71.

This paper gives a rigorous derivation of a functional proposed by Aftalion and Rivière [Phys. Rev. A 64 (2001) 043611] to characterize the energy of vortex filaments in a rotationally forced Bose-Einstein condensate. This functional is derived as a $\Gamma$-limit of scaled versions of the Gross-Pitaevsky functional for the wave function of such a condensate. In most situations, the vortex filament energy functional is either unbounded below or has only trivial minimizers, but we establish the existence of large numbers of nontrivial local minimizers and we prove that, given any such local minimizer, the Gross-Pitaevsky functional has a local minimizer that is nearby (in a suitable sense) whenever a scaling parameter is sufficiently small.

Classification : 35Q40, 35B25, 49Q20
Mots clés : Gross-Pitaevsky, vortices, gamma-convergence, Thomas-Fermi limit, rectifiable currents

[1] A. Aftalion and R.L. Jerrard, On the shape of vortices for a rotating Bose-Einstein condensate. Phys. Rev. A 66 (2002) 023611.

[2] A. Aftalion and R. L. Jerrard, Properties of a single vortex solution in a rotating Bose-Einstein condensate. C. R. Acad. Sci. Paris Ser. I 336 (2003) 713-718. | Zbl

[3] A. Aftalion and T. Rivière, Vortex energy and vortex bending for a rotating Bose-Einstein condensate. Phys. Rev. A 64 (2001) 043611.

[4] G. Alberti, S. Baldo and G. Orlandi, Functions with prescribed singularities. J. Eur. Math. Soc. 5 (2003) 275-311. | Zbl

[5] G. Alberti, S. Baldo and G. Orlandi, Variational convergence for functionals of Ginzburg-Landau type. Indiana Univ. Math J. 54 (2005) 1411-1472.

[6] N. Andre and I. Shafrir, Asymptotic behavior of minimizers for the Ginzburg-Landau functional with weight. I, II. Arch. Rational Mech. Anal. 142 (1998) 45-73, 75-98. | Zbl

[7] F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices. Birkhauser, New-York (1994). | MR | Zbl

[8] H. Brezis, J.M. Coron, and E.H. Lieb, Harmonic maps with defects. Comm. Math. Phys. 107 (1986) 649-705. | Zbl

[9] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, London (1992). | MR | Zbl

[10] H. Federer, Geometric Measure Theory. Springer-Verlag, Berlin (1969). | MR | Zbl

[11] M. Giaquinta, G. Modica and J. Soucek, Cartesian Currents in the Calculus of Variations. I, II. Springer-Verlag, New York (1998). | MR | Zbl

[12] R.L. Jerrard and H.M. Soner, The Jacobian and the Ginzburg-Landau functional. Cal. Var. 14 (2002) 151-191. | Zbl

[13] R.L. Jerrard, A. Montero, and P. Sternberg, Local minimizers of the Ginzburg-Landau energy with magnetic field in three dimensions. Comm. Math. Phys. 249 (2004) 549-577. | Zbl

[14] R.V. Kohn and P. Sternberg, Local minimizers and singular perturbations. Proc. Royal Soc. Edin. 111A (1989) 69-84. | Zbl

[15] L. Lassoued and P. Mironescu, Ginzburg-Landau type energy with discontinuous constraint. J. Anal. Math. 77 (1999) 1-26. | Zbl

[16] A. Montero, P. Sternberg, and W. Ziemer, Local minimizers with vortices to the Ginzburg-Landau system in 3-d. Comm. Pure Appl. Math 57 (2004) 99-125. | Zbl

[17] C. Raman, J. R. Abo-Shaeer, J. M. Vogels, K. Xu and W. Ketterle, Vortex nucleation in a stirred Bose-Einstein condensate. Phys. Rev. Lett. 87 (2001) 210402.

[18] T. Rivière, Line vortices in the $U\left(1\right)$-Higgs model. Cont. Opt. Calc. Var. 1 (1996) 77-167. | Numdam | Zbl

[19] P. Rosenbuch, V. Bretin, and J. Dalibard, Dynamics of a single vortex line in a Bose-Einstein condensate. Phys. Rev. Lett. 89 (2002) 200403.

[20] E. Sandier and S. Serfaty. A product estimate for Ginzburg-Landau and corollaries. J. Funct. Anal. 211 (2004) 219-244. | Zbl