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May  2015, 35(5): 1933-1968. doi: 10.3934/dcds.2015.35.1933

Topological defects in the abelian Higgs model

Magdalena Czubak 1, and  Robert L. Jerrard 2,

1. 

Department of Mathematical Sciences, Binghamton University (SUNY), Binghamton, NY 13902-6000
2. 

Department of Mathematics, University of Toronto, Bahen Centre 40 St. George St., Room 6290, Toronto, ON M5S 2E4, Canada

Received  December 2013 Revised  September 2014 Published  December 2014

We give a rigorous description of the dynamics of the Nielsen-Olesen vortex line. In particular, given a worldsheet of a string, we construct initial data such that the corresponding solution of the abelian Higgs model will concentrate near the evolution of the string. Moreover, the constructed solution stays close to the Nielsen-Olesen vortex solution.
Keywords: Abelian Higgs model, minimizers., vortices, Nambu-Goto action, cosmic strings, timelike minimal surface, Nielsen-Olesen vortex line.
Mathematics Subject Classification: Primary: 35L70, 49Q05; Secondary: 35Q85, 35B4.
Citation: Magdalena Czubak, Robert L. Jerrard. Topological defects in the abelian Higgs model. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1933-1968. doi: 10.3934/dcds.2015.35.1933
References:
[1]

Y. Almog, L. Berlyand, D. Golovaty and I. Shafrir, Global minimizers for a $p$-Ginzburg-Landau-type energy in $\mathbb{R}^2$, J. Funct. Anal., 256 (2009), 2268-2290. doi: 10.1016/j.jfa.2008.09.020.

[2]

G. Bellettini, J. Hoppe, M. Novaga and G. Orlandi, Closure and convexity results for closed relativistic strings, Complex Anal. Oper. Theory, 4 (2010), 473-496. doi: 10.1007/s11785-010-0060-y.

[3]

G. Bellettini, M. Novaga and G. Orlandi, Time-like minimal submanifolds as singular limits of nonlinear wave equations, Phys. D, 239 (2010), 335-339. doi: 10.1016/j.physd.2009.12.004.

[4]

M. S. Berger and Y. Y. Chen, Symmetric vortices for the Ginzburg-Landau equations of superconductivity and the nonlinear desingularization phenomenon, J. Funct. Anal., 82 (1989), 259-295. doi: 10.1016/0022-1236(89)90071-2.

[5]

P. Goddard, From Dual Models to String Theory, The birth of string theory, Cambridge University Press, Cambridge, 2012. xxvi+636 pp.

[6]

T. Gotô, Relativistic quantum mechanics of one-dimensional mechanical continuum and subsidiary conditon of dual resonance model, Progr. Theoret. Phys., 46 (1971), 1560-1569. doi: 10.1143/PTP.46.1560.

[7]

S. Gustafson and I. M. Sigal, The stability of magnetic vortices, Comm. Math. Phys., 212 (2000), 257-275. doi: 10.1007/PL00005526.

[8]

S. Gustafson and I. M. Sigal, Effective dynamics of magnetic vortices, Adv. Math., 199 (2006), 448-498. doi: 10.1016/j.aim.2005.05.017.

[9]

A. Jaffe and C. Taubes, Vortices and Monopoles, vol. 2 of Progress in Physics, Birkhäuser Boston, Mass., 1980, Structure of static gauge theories.

[10]

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

[11]

R. L. Jerrard, Vortex dynamics for the Ginzburg-Landau wave equation, Calc. Var. Partial Differential Equations, 9 (1999), 1-30. doi: 10.1007/s005260050131.

[12]

R. Jerrard, Defects in semilinear wave equations and timelike minimal surfaces in Minkowski space, Anal. PDE, 4 (2011), 285-340. doi: 10.2140/apde.2011.4.285.

[13]

R. Jerrard, M. Novaga and G. Orlandi, On the regularity of timelike extremal surfaces, To appear, Commun. Comtemp. Math., arXiv:1211.3162. doi: 10.1142/S0219199714500485.

[14]

M. Keel, Global existence for critical power Yang-Mills-Higgs equations in $R^{3+1}$, Comm. Partial Differential Equations, 22 (1997), 1161-1225.

[15]

T. W. B. Kibble, Topology of cosmic domains and strings, Journal of Physics A: Mathematical and General, 9 (1976), 1387. doi: 10.1088/0305-4470/9/8/029.

[16]

S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy, Duke Math. J., 74 (1994), 19-44. doi: 10.1215/S0012-7094-94-07402-4.

[17]

F. H. Lin, Vortex dynamics for the nonlinear wave equation, Comm. Pure Appl. Math., 52 (1999), 737-761. doi: 10.1002/(SICI)1097-0312(199906)52:6<737::AID-CPA3>3.0.CO;2-Y.

[18]

Y. Nambu, Duality and Hadrodynamics (Notes prepared for the Copenhagen High Energy Symposium, unpublished, 1970), Broken symmetry, vol. 13 of World Scientific Series in 20th Century Physics, World Scientific Publishing Co. Inc., River Edge, NJ, 1995, Selected papers of Y. Nambu, Edited and with a foreword by T. Eguchi and K. Nishijima.

[19]

L. Nguyen and G. Tian, On the smoothness of timelike maximal cylinders in three dimensional vacuum spacetimes, Classical Quantum Gravity, 30 (2013), 165010, 26 pp. doi: 10.1088/0264-9381/30/16/165010.

[20]

H. B. Nielsen and P. Olesen, Vortex-line models for dual strings, Nuclear Phys., 61 (1973), 45-61. doi: 10.1016/0550-3213(73)90350-7.

[21]

T. Rivière, Towards Jaffe and Taubes conjectures in the strongly repulsive limit, Manuscripta Math., 108 (2002), 217-273. doi: 10.1007/s002290200266.

[22]

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

[23]

S. Selberg and A. Tesfahun, Finite-energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge, Comm. Partial Differential Equations, 35 (2010), 1029-1057. doi: 10.1080/03605301003717100.

[24]

D. Stuart, Dynamics of abelian Higgs vortices in the near Bogomolny regime, Comm. Math. Phys., 159 (1994), 51-91, URL http://projecteuclid.org/getRecord?id=euclid.cmp/1104254491. doi: 10.1007/BF02100485.

[25]

D. M. A. Stuart, The geodesic hypothesis and non-topological solitons on pseudo-Riemannian manifolds, Ann. Sci. École Norm. Sup. (4), 37 (2004), 312-362. doi: 10.1016/j.ansens.2003.07.001.

[26]

A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1994.

[27]

Y. Yu, Vortex dynamics for the nonlinear Maxwell-Klein-Gordon equation, Arch. Ration. Mech. Anal., 201 (2011), 743-776. doi: 10.1007/s00205-011-0422-2.

show all references

References:
[1]

Y. Almog, L. Berlyand, D. Golovaty and I. Shafrir, Global minimizers for a $p$-Ginzburg-Landau-type energy in $\mathbb{R}^2$, J. Funct. Anal., 256 (2009), 2268-2290. doi: 10.1016/j.jfa.2008.09.020.

[2]

G. Bellettini, J. Hoppe, M. Novaga and G. Orlandi, Closure and convexity results for closed relativistic strings, Complex Anal. Oper. Theory, 4 (2010), 473-496. doi: 10.1007/s11785-010-0060-y.

[3]

G. Bellettini, M. Novaga and G. Orlandi, Time-like minimal submanifolds as singular limits of nonlinear wave equations, Phys. D, 239 (2010), 335-339. doi: 10.1016/j.physd.2009.12.004.

[4]

M. S. Berger and Y. Y. Chen, Symmetric vortices for the Ginzburg-Landau equations of superconductivity and the nonlinear desingularization phenomenon, J. Funct. Anal., 82 (1989), 259-295. doi: 10.1016/0022-1236(89)90071-2.

[5]

P. Goddard, From Dual Models to String Theory, The birth of string theory, Cambridge University Press, Cambridge, 2012. xxvi+636 pp.

[6]

T. Gotô, Relativistic quantum mechanics of one-dimensional mechanical continuum and subsidiary conditon of dual resonance model, Progr. Theoret. Phys., 46 (1971), 1560-1569. doi: 10.1143/PTP.46.1560.

[7]

S. Gustafson and I. M. Sigal, The stability of magnetic vortices, Comm. Math. Phys., 212 (2000), 257-275. doi: 10.1007/PL00005526.

[8]

S. Gustafson and I. M. Sigal, Effective dynamics of magnetic vortices, Adv. Math., 199 (2006), 448-498. doi: 10.1016/j.aim.2005.05.017.

[9]

A. Jaffe and C. Taubes, Vortices and Monopoles, vol. 2 of Progress in Physics, Birkhäuser Boston, Mass., 1980, Structure of static gauge theories.

[10]

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

[11]

R. L. Jerrard, Vortex dynamics for the Ginzburg-Landau wave equation, Calc. Var. Partial Differential Equations, 9 (1999), 1-30. doi: 10.1007/s005260050131.

[12]

R. Jerrard, Defects in semilinear wave equations and timelike minimal surfaces in Minkowski space, Anal. PDE, 4 (2011), 285-340. doi: 10.2140/apde.2011.4.285.

[13]

R. Jerrard, M. Novaga and G. Orlandi, On the regularity of timelike extremal surfaces, To appear, Commun. Comtemp. Math., arXiv:1211.3162. doi: 10.1142/S0219199714500485.

[14]

M. Keel, Global existence for critical power Yang-Mills-Higgs equations in $R^{3+1}$, Comm. Partial Differential Equations, 22 (1997), 1161-1225.

[15]

T. W. B. Kibble, Topology of cosmic domains and strings, Journal of Physics A: Mathematical and General, 9 (1976), 1387. doi: 10.1088/0305-4470/9/8/029.

[16]

S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy, Duke Math. J., 74 (1994), 19-44. doi: 10.1215/S0012-7094-94-07402-4.

[17]

F. H. Lin, Vortex dynamics for the nonlinear wave equation, Comm. Pure Appl. Math., 52 (1999), 737-761. doi: 10.1002/(SICI)1097-0312(199906)52:6<737::AID-CPA3>3.0.CO;2-Y.

[18]

Y. Nambu, Duality and Hadrodynamics (Notes prepared for the Copenhagen High Energy Symposium, unpublished, 1970), Broken symmetry, vol. 13 of World Scientific Series in 20th Century Physics, World Scientific Publishing Co. Inc., River Edge, NJ, 1995, Selected papers of Y. Nambu, Edited and with a foreword by T. Eguchi and K. Nishijima.

[19]

L. Nguyen and G. Tian, On the smoothness of timelike maximal cylinders in three dimensional vacuum spacetimes, Classical Quantum Gravity, 30 (2013), 165010, 26 pp. doi: 10.1088/0264-9381/30/16/165010.

[20]

H. B. Nielsen and P. Olesen, Vortex-line models for dual strings, Nuclear Phys., 61 (1973), 45-61. doi: 10.1016/0550-3213(73)90350-7.

[21]

T. Rivière, Towards Jaffe and Taubes conjectures in the strongly repulsive limit, Manuscripta Math., 108 (2002), 217-273. doi: 10.1007/s002290200266.

[22]

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

[23]

S. Selberg and A. Tesfahun, Finite-energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge, Comm. Partial Differential Equations, 35 (2010), 1029-1057. doi: 10.1080/03605301003717100.

[24]

D. Stuart, Dynamics of abelian Higgs vortices in the near Bogomolny regime, Comm. Math. Phys., 159 (1994), 51-91, URL http://projecteuclid.org/getRecord?id=euclid.cmp/1104254491. doi: 10.1007/BF02100485.

[25]

D. M. A. Stuart, The geodesic hypothesis and non-topological solitons on pseudo-Riemannian manifolds, Ann. Sci. École Norm. Sup. (4), 37 (2004), 312-362. doi: 10.1016/j.ansens.2003.07.001.

[26]

A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1994.

[27]

Y. Yu, Vortex dynamics for the nonlinear Maxwell-Klein-Gordon equation, Arch. Ration. Mech. Anal., 201 (2011), 743-776. doi: 10.1007/s00205-011-0422-2.

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