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Dynamics of Ginzburg-Landau and Gross-Pitaevskii vortices on manifolds
1. | Department of Mathematics, Indiana University, Bloomington, IN 47405, United States, United States |
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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