May  2014, 34(5): 1905-1931. doi: 10.3934/dcds.2014.34.1905

Dynamics of Ginzburg-Landau and Gross-Pitaevskii vortices on manifolds

1. 

Department of Mathematics, Indiana University, Bloomington, IN 47405, United States, United States

Received  November 2012 Revised  June 2013 Published  October 2013

We consider the dissipative heat flow and conservative Gross-Pitaevskii dynamics associated with the Ginzburg-Landau energy \begin{equation*} E_\varepsilon(u) = \int_{\mathcal M} \frac{|\nabla_g u|^2}{2} + \frac{(1-|u|^2)^2}{4\varepsilon^2} dv_g \end{equation*} posed on a Riemannian $2$-manifold $\mathcal{M}$ endowed with a metric $g$. In the $ε \to 0$ limit, we show the vortices of the solutions to these two problems evolve according to the gradient flow and Hamiltonian point-vortex flow respectively, associated with the renormalized energy on $\mathcal{M}.$ For the heat flow, we then specialize to the case where $\mathcal{M}=S^2$ and study the limiting system of ODE's and establish an annihilation result. Finally, for the Ginzburg-Landau heat flow on $S^2$, we derive some weighted energy identities.
Citation: Ko-Shin Chen, Peter Sternberg. Dynamics of Ginzburg-Landau and Gross-Pitaevskii vortices on manifolds. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1905-1931. doi: 10.3934/dcds.2014.34.1905
References:
[1]

S. Baraket, Critical points of the Ginzburg-Landau system on a Riemannian surface, Asymptotic Analysisl, 13 (1996), 277-317.

[2]

P. Bauman, C. Chen, D. Phillips and P. Sternberg, Vortex annihilation in nonlinear heat flow for Ginzburg-Landau systems, Euro. J. Applied Math., 6 (1995), 115-126. doi: 10.1017/S0956792500001728.

[3]

F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices, Birkhäuser, Boston, 2004.

[4]

F. Bethuel, G. Orlandi and D. Smets, Collisions and phase-vortex interactions in dissipative Ginzburg-Landau dynamics, Duke Math. J., 130 (2005), 523-614. doi: 10.1215/S0012-7094-05-13034-4.

[5]

F. Bethuel, G. Orlandi and D. Smets, Quantization and motion law for Ginzburg-Landau vortices, Arch. Ration. Mech. Anal., 183 (2007), 315-370. doi: 10.1007/s00205-006-0018-4.

[6]

F. Bethuel, G. Orlandi and D. Smets, Dynamics of multiple degree Ginzburg-Landau vortices, Comm. Math. Phys., 272 (2007), 229-261. doi: 10.1007/s00220-007-0206-6.

[7]

N. Burq, P. Gérard and N. Tzvetkov, Stricharz, Inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math., 126 (2004), 569-605. doi: 10.1353/ajm.2004.0016.

[8]

K. Chen, Instability of Ginzburg-Landau Vortices on Manifolds, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 143 (2013), 337-350. doi: 10.1017/S0308210511000795.

[9]

A. Contreras, On the first critical field in Ginzburg-Landau theory for thin shells and manifolds, Arch. Rat. Mech. Anal., 200, (2011), 563-611. doi: 10.1007/s00205-010-0352-4.

[10]

A. Contreras and P. Sternberg, Gamma-convergence and the emergence of vortices for Ginzburg-Landau on thin shells and manifolds, Calc. Var. Partial Differential Equations, 38 (2010), 243-274. doi: 10.1007/s00526-009-0285-7.

[11]

J. E. Colliander and R. L. Jerrard, Ginzburg-Landau vortices: Weak stability and Schrödinger equation dynamics, Inter. Math. Res. Notices, 7 (1998), 333-358. doi: 10.1155/S1073792898000221.

[12]

J. E. Colliander and R. L. Jerrard, Ginzburg-Landau vortices: Weak stability and Schrödinger equation dynamics, Journal d'Analyse Mathematique, 77 (1999), 129-205. doi: 10.1007/BF02791260.

[13]

M. Gelantalis and P. Sternberg, Rotating $2N$-vortex solutions to Gross-Pitaevskii on $S^2$, J. Math. Phys., 53 (2012), 083701. doi: 10.1063/1.4739748.

[14]

V. Ginzburg and L. Landau, On the theory of superconductivity, Zh. Eksper. Teoret. Fiz., 20 (1950), 1064-1082.

[15]

R. L. Jerrard, Lower bounds for generalized Ginzburg-Landau functionals, SIAM J. Math Anal., 30 (1999), 721-746. doi: 10.1137/S0036141097300581.

[16]

R. L. Jerrard and H. M. Soner, Dynamics of Ginzburg-Landau vortices, Arch. Rat. Mech. Anal., 142 (1998), 99-125. doi: 10.1007/s002050050085.

[17]

R. L. Jerrard and H. M. Soner, The Jacobian and the Ginzburg-Landau energy, Calc. Var. Partial Differential Equations, 14 (2002), 151-191. doi: 10.1007/s005260100093.

[18]

R. L. Jerrard and D. Spirn, Refined Jacobian estimates and Gross-Pitaevsky vortex dynamics, Arch. Rat. Mech. Anal., 190 (2008), 425-475. doi: 10.1007/s00205-008-0167-8.

[19]

F.-H. Lin, Some dynamical properties of Ginzburg-Landau vortices, Comm. Pure Appl. Math., 49 (1996), 323-359. doi: 10.1002/(SICI)1097-0312(199604)49:4<323::AID-CPA1>3.0.CO;2-E.

[20]

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), 249-274. doi: 10.1007/s002200050529.

[21]

P. K. Newton, The N-Vortex Problem- Analytical Techniques, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4684-9290-3.

[22]

P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, 171. Springer-Verlag, New York, 1998.

[23]

J. Rubinstein and P. Sternberg, On the slow motion of vortices in the Ginzburg-Landau heat flow, SIAM J. Math. Anal., 26 (1995), 1452-1466. doi: 10.1137/S0036141093259403.

[24]

E. Sandier, Lower bounds for the energy of unit vector fields and applications, J. Funct. Anal., 152 (1998), 379-403. doi: 10.1006/jfan.1997.3170.

[25]

E. Sandier and S. Serfaty, Vortices in the Magnetic Ginzburg-Landau Mode, Progress in Nonlinear Differential Equations and their Applications, 70. Birkhäuser Boston, Inc., Boston, MA, 2007.

show all references

References:
[1]

S. Baraket, Critical points of the Ginzburg-Landau system on a Riemannian surface, Asymptotic Analysisl, 13 (1996), 277-317.

[2]

P. Bauman, C. Chen, D. Phillips and P. Sternberg, Vortex annihilation in nonlinear heat flow for Ginzburg-Landau systems, Euro. J. Applied Math., 6 (1995), 115-126. doi: 10.1017/S0956792500001728.

[3]

F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices, Birkhäuser, Boston, 2004.

[4]

F. Bethuel, G. Orlandi and D. Smets, Collisions and phase-vortex interactions in dissipative Ginzburg-Landau dynamics, Duke Math. J., 130 (2005), 523-614. doi: 10.1215/S0012-7094-05-13034-4.

[5]

F. Bethuel, G. Orlandi and D. Smets, Quantization and motion law for Ginzburg-Landau vortices, Arch. Ration. Mech. Anal., 183 (2007), 315-370. doi: 10.1007/s00205-006-0018-4.

[6]

F. Bethuel, G. Orlandi and D. Smets, Dynamics of multiple degree Ginzburg-Landau vortices, Comm. Math. Phys., 272 (2007), 229-261. doi: 10.1007/s00220-007-0206-6.

[7]

N. Burq, P. Gérard and N. Tzvetkov, Stricharz, Inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math., 126 (2004), 569-605. doi: 10.1353/ajm.2004.0016.

[8]

K. Chen, Instability of Ginzburg-Landau Vortices on Manifolds, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 143 (2013), 337-350. doi: 10.1017/S0308210511000795.

[9]

A. Contreras, On the first critical field in Ginzburg-Landau theory for thin shells and manifolds, Arch. Rat. Mech. Anal., 200, (2011), 563-611. doi: 10.1007/s00205-010-0352-4.

[10]

A. Contreras and P. Sternberg, Gamma-convergence and the emergence of vortices for Ginzburg-Landau on thin shells and manifolds, Calc. Var. Partial Differential Equations, 38 (2010), 243-274. doi: 10.1007/s00526-009-0285-7.

[11]

J. E. Colliander and R. L. Jerrard, Ginzburg-Landau vortices: Weak stability and Schrödinger equation dynamics, Inter. Math. Res. Notices, 7 (1998), 333-358. doi: 10.1155/S1073792898000221.

[12]

J. E. Colliander and R. L. Jerrard, Ginzburg-Landau vortices: Weak stability and Schrödinger equation dynamics, Journal d'Analyse Mathematique, 77 (1999), 129-205. doi: 10.1007/BF02791260.

[13]

M. Gelantalis and P. Sternberg, Rotating $2N$-vortex solutions to Gross-Pitaevskii on $S^2$, J. Math. Phys., 53 (2012), 083701. doi: 10.1063/1.4739748.

[14]

V. Ginzburg and L. Landau, On the theory of superconductivity, Zh. Eksper. Teoret. Fiz., 20 (1950), 1064-1082.

[15]

R. L. Jerrard, Lower bounds for generalized Ginzburg-Landau functionals, SIAM J. Math Anal., 30 (1999), 721-746. doi: 10.1137/S0036141097300581.

[16]

R. L. Jerrard and H. M. Soner, Dynamics of Ginzburg-Landau vortices, Arch. Rat. Mech. Anal., 142 (1998), 99-125. doi: 10.1007/s002050050085.

[17]

R. L. Jerrard and H. M. Soner, The Jacobian and the Ginzburg-Landau energy, Calc. Var. Partial Differential Equations, 14 (2002), 151-191. doi: 10.1007/s005260100093.

[18]

R. L. Jerrard and D. Spirn, Refined Jacobian estimates and Gross-Pitaevsky vortex dynamics, Arch. Rat. Mech. Anal., 190 (2008), 425-475. doi: 10.1007/s00205-008-0167-8.

[19]

F.-H. Lin, Some dynamical properties of Ginzburg-Landau vortices, Comm. Pure Appl. Math., 49 (1996), 323-359. doi: 10.1002/(SICI)1097-0312(199604)49:4<323::AID-CPA1>3.0.CO;2-E.

[20]

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), 249-274. doi: 10.1007/s002200050529.

[21]

P. K. Newton, The N-Vortex Problem- Analytical Techniques, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4684-9290-3.

[22]

P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, 171. Springer-Verlag, New York, 1998.

[23]

J. Rubinstein and P. Sternberg, On the slow motion of vortices in the Ginzburg-Landau heat flow, SIAM J. Math. Anal., 26 (1995), 1452-1466. doi: 10.1137/S0036141093259403.

[24]

E. Sandier, Lower bounds for the energy of unit vector fields and applications, J. Funct. Anal., 152 (1998), 379-403. doi: 10.1006/jfan.1997.3170.

[25]

E. Sandier and S. Serfaty, Vortices in the Magnetic Ginzburg-Landau Mode, Progress in Nonlinear Differential Equations and their Applications, 70. Birkhäuser Boston, Inc., Boston, MA, 2007.

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