American Institute of Mathematical Sciences

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

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

Ko-Shin Chen 1, and  Peter Sternberg 1,

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.
Keywords: Vortex motion, Gross-Pitaevskii., Ginzburg-Landau.
Mathematics Subject Classification: Primary: 35R01, 35K1.
Citation: Ko-Shin Chen, Peter Sternberg. Dynamics of Ginzburg-Landau and Gross-Pitaevskii vortices on manifolds. Discrete & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Hans G. Kaper, Peter Takáč. Bifurcating vortex solutions of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 871-880. doi: 10.3934/dcds.1999.5.871
[2]

Jun Yang. Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2359-2388. doi: 10.3934/dcds.2014.34.2359
[3]

Shijin Ding, Qiang Du. The global minimizers and vortex solutions to a Ginzburg-Landau model of superconducting films. Communications on Pure & Applied Analysis, 2002, 1 (3) : 327-340. doi: 10.3934/cpaa.2002.1.327
[4]

Lipeng Duan, Jun Yang. On the non-degeneracy of radial vortex solutions for a coupled Ginzburg-Landau system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021056
[5]

Hans G. Kaper, Bixiang Wang, Shouhong Wang. Determining nodes for the Ginzburg-Landau equations of superconductivity. Discrete & Continuous Dynamical Systems, 1998, 4 (2) : 205-224. doi: 10.3934/dcds.1998.4.205
[6]

Mickaël Dos Santos, Oleksandr Misiats. Ginzburg-Landau model with small pinning domains. Networks & Heterogeneous Media, 2011, 6 (4) : 715-753. doi: 10.3934/nhm.2011.6.715
[7]

Fanghua Lin, Ping Zhang. On the hydrodynamic limit of Ginzburg-Landau vortices. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 121-142. doi: 10.3934/dcds.2000.6.121
[8]

Leonid Berlyand, Volodymyr Rybalko, Nung Kwan Yip. Renormalized Ginzburg-Landau energy and location of near boundary vortices. Networks & Heterogeneous Media, 2012, 7 (1) : 179-196. doi: 10.3934/nhm.2012.7.179
[9]

Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345
[10]

Leonid Berlyand, Petru Mironescu. Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain. Networks & Heterogeneous Media, 2008, 3 (3) : 461-487. doi: 10.3934/nhm.2008.3.461
[11]

N. Maaroufi. Topological entropy by unit length for the Ginzburg-Landau equation on the line. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 647-662. doi: 10.3934/dcds.2014.34.647
[12]

Leonid Berlyand, Volodymyr Rybalko. Homogenized description of multiple Ginzburg-Landau vortices pinned by small holes. Networks & Heterogeneous Media, 2013, 8 (1) : 115-130. doi: 10.3934/nhm.2013.8.115
[13]

Kolade M. Owolabi, Edson Pindza. Numerical simulation of multidimensional nonlinear fractional Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 835-851. doi: 10.3934/dcdss.2020048
[14]

Dmitry Turaev, Sergey Zelik. Analytical proof of space-time chaos in Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1713-1751. doi: 10.3934/dcds.2010.28.1713
[15]

Satoshi Kosugi, Yoshihisa Morita. Phase pattern in a Ginzburg-Landau model with a discontinuous coefficient in a ring. Discrete & Continuous Dynamical Systems, 2006, 14 (1) : 149-168. doi: 10.3934/dcds.2006.14.149
[16]

Jingna Li, Li Xia. The Fractional Ginzburg-Landau equation with distributional initial data. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2173-2187. doi: 10.3934/cpaa.2013.12.2173
[17]

Noboru Okazawa, Tomomi Yokota. Smoothing effect for generalized complex Ginzburg-Landau equations in unbounded domains. Conference Publications, 2001, 2001 (Special) : 280-288. doi: 10.3934/proc.2001.2001.280
[18]

N. I. Karachalios, H. E. Nistazakis, A. N. Yannacopoulos. Remarks on the asymptotic behavior of solutions of complex discrete Ginzburg-Landau equations. Conference Publications, 2005, 2005 (Special) : 476-486. doi: 10.3934/proc.2005.2005.476
[19]

Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 665-682. doi: 10.3934/cpaa.2005.4.665
[20]

Yuta Kugo, Motohiro Sobajima, Toshiyuki Suzuki, Tomomi Yokota, Kentarou Yoshii. Solvability of a class of complex Ginzburg-Landau equations in periodic Sobolev spaces. Conference Publications, 2015, 2015 (special) : 754-763. doi: 10.3934/proc.2015.0754

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences