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

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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

Abstract
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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 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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