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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 - A, 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 $\mathbbR^2$,, J. Funct. Anal., 256 (2009), 2268.  doi: 10.1016/j.jfa.2008.09.020.  Google Scholar

[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.  doi: 10.1007/s11785-010-0060-y.  Google Scholar

[3]

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

[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.  doi: 10.1016/0022-1236(89)90071-2.  Google Scholar

[5]

P. Goddard, From Dual Models to String Theory,, The birth of string theory, (2012).   Google Scholar

[6]

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

[7]

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

[8]

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

[9]

A. Jaffe and C. Taubes, Vortices and Monopoles, vol. 2 of Progress in Physics,, Birkhäuser Boston, (1980).   Google Scholar

[10]

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

[11]

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

[12]

R. Jerrard, Defects in semilinear wave equations and timelike minimal surfaces in Minkowski space,, Anal. PDE, 4 (2011), 285.  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]

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

[15]

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

[16]

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

[17]

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

[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., (1995).   Google Scholar

[19]

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

[20]

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

[21]

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

[22]

E. Sandier and S. Serfaty, Vortices in the Magnetic Ginzburg-Landau Model,, Progress in Nonlinear Differential Equations and their Applications, (2007).   Google Scholar

[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.  doi: 10.1080/03605301003717100.  Google Scholar

[24]

D. Stuart, Dynamics of abelian Higgs vortices in the near Bogomolny regime,, Comm. Math. Phys., 159 (1994), 51.  doi: 10.1007/BF02100485.  Google Scholar

[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.  doi: 10.1016/j.ansens.2003.07.001.  Google Scholar

[26]

A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects,, Cambridge Monographs on Mathematical Physics, (1994).   Google Scholar

[27]

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

show all references

References:
[1]

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

[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.  doi: 10.1007/s11785-010-0060-y.  Google Scholar

[3]

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

[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.  doi: 10.1016/0022-1236(89)90071-2.  Google Scholar

[5]

P. Goddard, From Dual Models to String Theory,, The birth of string theory, (2012).   Google Scholar

[6]

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

[7]

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

[8]

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

[9]

A. Jaffe and C. Taubes, Vortices and Monopoles, vol. 2 of Progress in Physics,, Birkhäuser Boston, (1980).   Google Scholar

[10]

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

[11]

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

[12]

R. Jerrard, Defects in semilinear wave equations and timelike minimal surfaces in Minkowski space,, Anal. PDE, 4 (2011), 285.  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]

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

[15]

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

[16]

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

[17]

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

[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., (1995).   Google Scholar

[19]

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

[20]

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

[21]

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

[22]

E. Sandier and S. Serfaty, Vortices in the Magnetic Ginzburg-Landau Model,, Progress in Nonlinear Differential Equations and their Applications, (2007).   Google Scholar

[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.  doi: 10.1080/03605301003717100.  Google Scholar

[24]

D. Stuart, Dynamics of abelian Higgs vortices in the near Bogomolny regime,, Comm. Math. Phys., 159 (1994), 51.  doi: 10.1007/BF02100485.  Google Scholar

[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.  doi: 10.1016/j.ansens.2003.07.001.  Google Scholar

[26]

A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects,, Cambridge Monographs on Mathematical Physics, (1994).   Google Scholar

[27]

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

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