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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 & Continuous Dynamical Systems, 2015, 35 (5) : 1933-1968. doi: 10.3934/dcds.2015.35.1933
References:
[1]

J. Funct. Anal., 256 (2009), 2268-2290. doi: 10.1016/j.jfa.2008.09.020.  Google Scholar

[2]

Complex Anal. Oper. Theory, 4 (2010), 473-496. doi: 10.1007/s11785-010-0060-y.  Google Scholar

[3]

Phys. D, 239 (2010), 335-339. doi: 10.1016/j.physd.2009.12.004.  Google Scholar

[4]

J. Funct. Anal., 82 (1989), 259-295. doi: 10.1016/0022-1236(89)90071-2.  Google Scholar

[5]

The birth of string theory, Cambridge University Press, Cambridge, 2012. xxvi+636 pp.  Google Scholar

[6]

Progr. Theoret. Phys., 46 (1971), 1560-1569. doi: 10.1143/PTP.46.1560.  Google Scholar

[7]

Comm. Math. Phys., 212 (2000), 257-275. doi: 10.1007/PL00005526.  Google Scholar

[8]

Adv. Math., 199 (2006), 448-498. doi: 10.1016/j.aim.2005.05.017.  Google Scholar

[9]

Birkhäuser Boston, Mass., 1980, Structure of static gauge theories.  Google Scholar

[10]

SIAM J. Math. Anal., 30 (1999), 721-746 (electronic). doi: 10.1137/S0036141097300581.  Google Scholar

[11]

Calc. Var. Partial Differential Equations, 9 (1999), 1-30. doi: 10.1007/s005260050131.  Google Scholar

[12]

Anal. PDE, 4 (2011), 285-340. doi: 10.2140/apde.2011.4.285.  Google Scholar

[13]

R. Jerrard, M. Novaga and G. Orlandi, On the regularity of timelike extremal surfaces,, To appear, ().  doi: 10.1142/S0219199714500485.  Google Scholar

[14]

Comm. Partial Differential Equations, 22 (1997), 1161-1225.  Google Scholar

[15]

Journal of Physics A: Mathematical and General, 9 (1976), 1387. doi: 10.1088/0305-4470/9/8/029.  Google Scholar

[16]

Duke Math. J., 74 (1994), 19-44. doi: 10.1215/S0012-7094-94-07402-4.  Google Scholar

[17]

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.  Google Scholar

[18]

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.  Google Scholar

[19]

Classical Quantum Gravity, 30 (2013), 165010, 26 pp. doi: 10.1088/0264-9381/30/16/165010.  Google Scholar

[20]

Nuclear Phys., 61 (1973), 45-61. doi: 10.1016/0550-3213(73)90350-7.  Google Scholar

[21]

Manuscripta Math., 108 (2002), 217-273. doi: 10.1007/s002290200266.  Google Scholar

[22]

Progress in Nonlinear Differential Equations and their Applications, 70, Birkhäuser Boston Inc., Boston, MA, 2007.  Google Scholar

[23]

Comm. Partial Differential Equations, 35 (2010), 1029-1057. doi: 10.1080/03605301003717100.  Google Scholar

[24]

Comm. Math. Phys., 159 (1994), 51-91, URL http://projecteuclid.org/getRecord?id=euclid.cmp/1104254491. doi: 10.1007/BF02100485.  Google Scholar

[25]

Ann. Sci. École Norm. Sup. (4), 37 (2004), 312-362. doi: 10.1016/j.ansens.2003.07.001.  Google Scholar

[26]

Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1994.  Google Scholar

[27]

Arch. Ration. Mech. Anal., 201 (2011), 743-776. doi: 10.1007/s00205-011-0422-2.  Google Scholar

show all references

References:
[1]

J. Funct. Anal., 256 (2009), 2268-2290. doi: 10.1016/j.jfa.2008.09.020.  Google Scholar

[2]

Complex Anal. Oper. Theory, 4 (2010), 473-496. doi: 10.1007/s11785-010-0060-y.  Google Scholar

[3]

Phys. D, 239 (2010), 335-339. doi: 10.1016/j.physd.2009.12.004.  Google Scholar

[4]

J. Funct. Anal., 82 (1989), 259-295. doi: 10.1016/0022-1236(89)90071-2.  Google Scholar

[5]

The birth of string theory, Cambridge University Press, Cambridge, 2012. xxvi+636 pp.  Google Scholar

[6]

Progr. Theoret. Phys., 46 (1971), 1560-1569. doi: 10.1143/PTP.46.1560.  Google Scholar

[7]

Comm. Math. Phys., 212 (2000), 257-275. doi: 10.1007/PL00005526.  Google Scholar

[8]

Adv. Math., 199 (2006), 448-498. doi: 10.1016/j.aim.2005.05.017.  Google Scholar

[9]

Birkhäuser Boston, Mass., 1980, Structure of static gauge theories.  Google Scholar

[10]

SIAM J. Math. Anal., 30 (1999), 721-746 (electronic). doi: 10.1137/S0036141097300581.  Google Scholar

[11]

Calc. Var. Partial Differential Equations, 9 (1999), 1-30. doi: 10.1007/s005260050131.  Google Scholar

[12]

Anal. PDE, 4 (2011), 285-340. doi: 10.2140/apde.2011.4.285.  Google Scholar

[13]

R. Jerrard, M. Novaga and G. Orlandi, On the regularity of timelike extremal surfaces,, To appear, ().  doi: 10.1142/S0219199714500485.  Google Scholar

[14]

Comm. Partial Differential Equations, 22 (1997), 1161-1225.  Google Scholar

[15]

Journal of Physics A: Mathematical and General, 9 (1976), 1387. doi: 10.1088/0305-4470/9/8/029.  Google Scholar

[16]

Duke Math. J., 74 (1994), 19-44. doi: 10.1215/S0012-7094-94-07402-4.  Google Scholar

[17]

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.  Google Scholar

[18]

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.  Google Scholar

[19]

Classical Quantum Gravity, 30 (2013), 165010, 26 pp. doi: 10.1088/0264-9381/30/16/165010.  Google Scholar

[20]

Nuclear Phys., 61 (1973), 45-61. doi: 10.1016/0550-3213(73)90350-7.  Google Scholar

[21]

Manuscripta Math., 108 (2002), 217-273. doi: 10.1007/s002290200266.  Google Scholar

[22]

Progress in Nonlinear Differential Equations and their Applications, 70, Birkhäuser Boston Inc., Boston, MA, 2007.  Google Scholar

[23]

Comm. Partial Differential Equations, 35 (2010), 1029-1057. doi: 10.1080/03605301003717100.  Google Scholar

[24]

Comm. Math. Phys., 159 (1994), 51-91, URL http://projecteuclid.org/getRecord?id=euclid.cmp/1104254491. doi: 10.1007/BF02100485.  Google Scholar

[25]

Ann. Sci. École Norm. Sup. (4), 37 (2004), 312-362. doi: 10.1016/j.ansens.2003.07.001.  Google Scholar

[26]

Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1994.  Google Scholar

[27]

Arch. Ration. Mech. Anal., 201 (2011), 743-776. doi: 10.1007/s00205-011-0422-2.  Google Scholar

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