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 & Continuous Dynamical Systems - A, 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. 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. doi: 10.1017/S0956792500001728. Google Scholar
[3]	F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices,, Birkhäuser, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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). 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. Google Scholar
[15]	R. L. Jerrard, Lower bounds for generalized Ginzburg-Landau functionals,, SIAM J. Math Anal., 30 (1999), 721. 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. 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. 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. 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. 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. doi: 10.1007/s002200050529. Google Scholar*
[21]	P. K. Newton, The N-Vortex Problem- Analytical Techniques,, Springer-Verlag, (2001). doi: 10.1007/978-1-4684-9290-3. Google Scholar
[22]	P. Petersen, Riemannian Geometry,, Graduate Texts in Mathematics, (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. 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. 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, (2007). Google Scholar

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References:

[1]	S. Baraket, Critical points of the Ginzburg-Landau system on a Riemannian surface,, Asymptotic Analysisl, 13 (1996), 277. 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. doi: 10.1017/S0956792500001728. Google Scholar
[3]	F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices,, Birkhäuser, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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). 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. Google Scholar
[15]	R. L. Jerrard, Lower bounds for generalized Ginzburg-Landau functionals,, SIAM J. Math Anal., 30 (1999), 721. 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. 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. 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. 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. 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. doi: 10.1007/s002200050529. Google Scholar*
[21]	P. K. Newton, The N-Vortex Problem- Analytical Techniques,, Springer-Verlag, (2001). doi: 10.1007/978-1-4684-9290-3. Google Scholar
[22]	P. Petersen, Riemannian Geometry,, Graduate Texts in Mathematics, (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. 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. 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, (2007). Google Scholar

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