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March  2018, 38(3): 1293-1314. doi: 10.3934/dcds.2018053

Topological solutions in the Maxwell-Chern-Simons model with anomalous magnetic moment

Youngae Lee

National Institute for Mathematical Sciences, Academic exchanges, KT Daeduk 2 Research Center, 70 Yuseong-daero 1689 beon-gil, Yuseong-gu, Daejeon, 34047, Republic of Korea

Received  March 2017 Revised  October 2017 Published  December 2017

In this paper, we consider a Maxwell-Chern-Simons model with anomalous magnetic moment. Our main goal is to show the existence and uniqueness of topological type solutions to this problem on a flat two torus for any configuration of vortex points. Moreover, we also discuss about the stability of topological solutions.

Keywords: Maxwell-Chern-Simons models, topological type solutions, stable solution, uniqueness.
Mathematics Subject Classification: 35J60.
Citation: Youngae Lee. Topological solutions in the Maxwell-Chern-Simons model with anomalous magnetic moment. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1293-1314. doi: 10.3934/dcds.2018053
References:
[1]

A. A. Abrikosov, On the magnetic properties of superconductors of the second group, Soviet Phys. JETP, 5 (1957), 1174-1182. 

[2]

D. Bartolucci, Y. Lee, C. S. Lin and M. Onodera, Asymptotic analysis of solutions to a gauged $O(3)$ sigma model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 651-685.  doi: 10.1016/j.anihpc.2014.03.001.

[3]

F. S. A. Cavalcante, M. S. Cunha and C. A. S. Almeida, Vortices in a nonminimal Maxwell-Chern-Simons O(3) sigma model, Phys. Lett. B, 475 (2000), 315-323.  doi: 10.1016/S0370-2693(00)00077-0.

[4]

D. Chae and O. Y. Imanuvilov, The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory, Commun. Math. Phys., 215 (2000), 119-142.  doi: 10.1007/s002200000302.

[5]

D. Chae and O. Y. Imanuvilov, Non-topological multivortex solutions to the self-dual Maxwell-Chern-Simons-Higgs systems, J. Funct. Anal., 196 (2002), 87-118.  doi: 10.1006/jfan.2002.3988.

[6]

H. Chan, C. C. Fu and C. S. Lin, Non-topological multivortex solutions to the self-dual Chern-Simons-Higgs equation, Commun. Math. Phys., 231 (2002), 189-221.  doi: 10.1007/s00220-002-0691-6.

[7]

X. Chen, S. Hastings, J. McLeod and Y. Yang, A nonlinear elliptic equation arising from gauge field theory and cosmology, Proc. R. Soc. Lond. A, 446 (1994), 453-478.  doi: 10.1098/rspa.1994.0115.

[8]

K. Choe, Self-dual non-topological vortices in a Maxwell-Chern-Simons model with non-minimal coupling, Lett. Math. Phys., 87 (2009), 47-65.  doi: 10.1007/s11005-009-0294-7.

[9]

K. Choe, Uniqueness of the topological multivortex solution in the selfdual Chern-Simons theory, J. Math. Phys., 46 (2005), 012305, 21pp.

[10]

K. Choe, J. Han, Y. Lee and C. S. Lin, Bubbling solutions for the Chern-Simons gauged $O(3)$ sigma model on a torus, Calc. Var. Partial Differential Equations, 54 (2015), 1275-1329.  doi: 10.1007/s00526-015-0825-2.

[11]

H. R. Christiansen, M. S. Cunha, J. A. Helayël-Neto, L. R. U. Manssur and A. L. M. A. Nogueira, $N =2$-Maxwell-Chern-Simons model with anomalous magnetic moment coupling via dimensional reduction, Int. J. Mod. Phys. A, 14 (1999), 147-159.  doi: 10.1142/S0217751X99000075.

[12]

H. R. Christiansen, M. S. Cunha, J. A. Helayël-Neto, L. R. U. Manssur and A. L. M. A. Nogueira, Self-dual vortices in a Maxwell-Chern-Simons model with non-minimal coupling, Int. J. Mod. Phys. A, 14 (1999), 1721-1735.  doi: 10.1142/S0217751X99000877.

[13]

W. Ding, J. Jost, J. Li, X. Peng and G. Wang, Self duality equations for Ginzburg-Landau and Seiberg-Witten type functionals with 6th order potentials, Comm. Math. Phys., 217 (2001), 383-407.  doi: 10.1007/s002200100377.

[14]

G. Dunne, Self-duality and Chern-Simons theories. Lecture Notes in Physics, Springer, Heidelberg, 1995.

[15]

Y. W. Fan, Y. Lee and C. S. Lin, Mixed type solutions of the SU(3) models on a torus, Comm. Math. Phys., 343 (2016), 233-271.  doi: 10.1007/s00220-015-2532-4.

[16]

D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[17]

J. Han and H. Huh, Self-dual vortices in a Maxwell-Chern-Simons model with non-minimal coupling, Lett. Math. Phys., 82 (2007), 9-24.  doi: 10.1007/s11005-007-0193-8.

[18]

J. Hong, P. Kim and P. Pac, Multivortex solutions of the abelian Chern-Simons-Higgs theory, Phys. Rev. Lett., 64 (1990), 2230-2233.  doi: 10.1103/PhysRevLett.64.2230.

[19]

G.'t Hooft, A property of electric and magnetic flux in nonabelian gauge theories, Nucl. Phys. B, 153 (1979), 141-160.  doi: 10.1016/0550-3213(79)90595-9.

[20]

R. Jackiw and E. Weinberg, Self-dual Chern-Simons vortices, Phys. Rev. Lett., 64 (1990), 2234-2237.  doi: 10.1103/PhysRevLett.64.2234.

[21] A. Jaffe and C. H. Taubes, Vortices and Monopoles, Birkh$ä$user, Boston, 1980. 
[22]

T. Lee and H. Min, Bogomol'nyi equations for solitons in Maxwell-Chern-Simons gauge theories with the magnetic moment interaction term, Phys. Rev. D, 50 (1994), 7738-7741.  doi: 10.1103/PhysRevD.50.7738.

[23]

C. Lee, K. Lee and H. Min, Self-dual Maxwell-Chern-Simons solitons, Phys. Lett. B, 252 (1990), 79-83.  doi: 10.1016/0370-2693(90)91084-O.

[24]

B. H. Lee, C. Lee and H. Min, Supersymmetric Chern-Simons vortex systems and fermion zero modes, Phys. Rev. D, 45 (1992), 4588-4599.  doi: 10.1103/PhysRevD.45.4588.

[25]

P. Navrátil, $N =2$ supersymmetry in a Chern-Simons system with the magnetic moment interaction, Phys. Lett. B, 365 (1996), 119-124.  doi: 10.1016/0370-2693(95)01254-0.

[26]

H. B. Nielsen and P. Olesen, Vortex-line models for dual-strings, Nucl. Phys. B, 61 (1973), 45-61. 

[27]

S. Paul and A. Khare, Charged vortices in an abelian Higgs model with Chern-Simons term, Phys. Lett. B, 174 (1986), 420-422.  doi: 10.1016/0370-2693(86)91028-2.

[28]

J. Spruck and Y. Yang, The existence of non-topological solitons in the self-dual Chern-Simons theory, Commun. Math. Phys., 149 (1992), 361-376.  doi: 10.1007/BF02097630.

[29]

G. Tarantello, Selfdual Maxwell-Chern-Simons vortices, Milan J. Math., 72 (2004), 29-80.  doi: 10.1007/s00032-004-0030-9.

[30]

G. Tarantello, Uniqueness of self-dual periodic Chern-Simons vortices of topological-type, Calc. Var. P.D.E., 29 (2007), 191-217.  doi: 10.1007/s00526-006-0062-9.

[31]

G. Tarantello, Selfdual gauge field vortices an analytical approach, In Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser, Boston, 2008.

[32]

M. Torres, Bogomol'nyi limit for nontopological solitons in a Chern-Simons model with anomalous magnetic moment, Phys. Rev. D, 46 (1992), 2295-2298.  doi: 10.1103/PhysRevD.46.R2295.

[33]

Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer Monograph in Mathematics. Springer, New York, 2001.

show all references

References:
[1]

A. A. Abrikosov, On the magnetic properties of superconductors of the second group, Soviet Phys. JETP, 5 (1957), 1174-1182. 

[2]

D. Bartolucci, Y. Lee, C. S. Lin and M. Onodera, Asymptotic analysis of solutions to a gauged $O(3)$ sigma model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 651-685.  doi: 10.1016/j.anihpc.2014.03.001.

[3]

F. S. A. Cavalcante, M. S. Cunha and C. A. S. Almeida, Vortices in a nonminimal Maxwell-Chern-Simons O(3) sigma model, Phys. Lett. B, 475 (2000), 315-323.  doi: 10.1016/S0370-2693(00)00077-0.

[4]

D. Chae and O. Y. Imanuvilov, The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory, Commun. Math. Phys., 215 (2000), 119-142.  doi: 10.1007/s002200000302.

[5]

D. Chae and O. Y. Imanuvilov, Non-topological multivortex solutions to the self-dual Maxwell-Chern-Simons-Higgs systems, J. Funct. Anal., 196 (2002), 87-118.  doi: 10.1006/jfan.2002.3988.

[6]

H. Chan, C. C. Fu and C. S. Lin, Non-topological multivortex solutions to the self-dual Chern-Simons-Higgs equation, Commun. Math. Phys., 231 (2002), 189-221.  doi: 10.1007/s00220-002-0691-6.

[7]

X. Chen, S. Hastings, J. McLeod and Y. Yang, A nonlinear elliptic equation arising from gauge field theory and cosmology, Proc. R. Soc. Lond. A, 446 (1994), 453-478.  doi: 10.1098/rspa.1994.0115.

[8]

K. Choe, Self-dual non-topological vortices in a Maxwell-Chern-Simons model with non-minimal coupling, Lett. Math. Phys., 87 (2009), 47-65.  doi: 10.1007/s11005-009-0294-7.

[9]

K. Choe, Uniqueness of the topological multivortex solution in the selfdual Chern-Simons theory, J. Math. Phys., 46 (2005), 012305, 21pp.

[10]

K. Choe, J. Han, Y. Lee and C. S. Lin, Bubbling solutions for the Chern-Simons gauged $O(3)$ sigma model on a torus, Calc. Var. Partial Differential Equations, 54 (2015), 1275-1329.  doi: 10.1007/s00526-015-0825-2.

[11]

H. R. Christiansen, M. S. Cunha, J. A. Helayël-Neto, L. R. U. Manssur and A. L. M. A. Nogueira, $N =2$-Maxwell-Chern-Simons model with anomalous magnetic moment coupling via dimensional reduction, Int. J. Mod. Phys. A, 14 (1999), 147-159.  doi: 10.1142/S0217751X99000075.

[12]

H. R. Christiansen, M. S. Cunha, J. A. Helayël-Neto, L. R. U. Manssur and A. L. M. A. Nogueira, Self-dual vortices in a Maxwell-Chern-Simons model with non-minimal coupling, Int. J. Mod. Phys. A, 14 (1999), 1721-1735.  doi: 10.1142/S0217751X99000877.

[13]

W. Ding, J. Jost, J. Li, X. Peng and G. Wang, Self duality equations for Ginzburg-Landau and Seiberg-Witten type functionals with 6th order potentials, Comm. Math. Phys., 217 (2001), 383-407.  doi: 10.1007/s002200100377.

[14]

G. Dunne, Self-duality and Chern-Simons theories. Lecture Notes in Physics, Springer, Heidelberg, 1995.

[15]

Y. W. Fan, Y. Lee and C. S. Lin, Mixed type solutions of the SU(3) models on a torus, Comm. Math. Phys., 343 (2016), 233-271.  doi: 10.1007/s00220-015-2532-4.

[16]

D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[17]

J. Han and H. Huh, Self-dual vortices in a Maxwell-Chern-Simons model with non-minimal coupling, Lett. Math. Phys., 82 (2007), 9-24.  doi: 10.1007/s11005-007-0193-8.

[18]

J. Hong, P. Kim and P. Pac, Multivortex solutions of the abelian Chern-Simons-Higgs theory, Phys. Rev. Lett., 64 (1990), 2230-2233.  doi: 10.1103/PhysRevLett.64.2230.

[19]

G.'t Hooft, A property of electric and magnetic flux in nonabelian gauge theories, Nucl. Phys. B, 153 (1979), 141-160.  doi: 10.1016/0550-3213(79)90595-9.

[20]

R. Jackiw and E. Weinberg, Self-dual Chern-Simons vortices, Phys. Rev. Lett., 64 (1990), 2234-2237.  doi: 10.1103/PhysRevLett.64.2234.

[21] A. Jaffe and C. H. Taubes, Vortices and Monopoles, Birkh$ä$user, Boston, 1980. 
[22]

T. Lee and H. Min, Bogomol'nyi equations for solitons in Maxwell-Chern-Simons gauge theories with the magnetic moment interaction term, Phys. Rev. D, 50 (1994), 7738-7741.  doi: 10.1103/PhysRevD.50.7738.

[23]

C. Lee, K. Lee and H. Min, Self-dual Maxwell-Chern-Simons solitons, Phys. Lett. B, 252 (1990), 79-83.  doi: 10.1016/0370-2693(90)91084-O.

[24]

B. H. Lee, C. Lee and H. Min, Supersymmetric Chern-Simons vortex systems and fermion zero modes, Phys. Rev. D, 45 (1992), 4588-4599.  doi: 10.1103/PhysRevD.45.4588.

[25]

P. Navrátil, $N =2$ supersymmetry in a Chern-Simons system with the magnetic moment interaction, Phys. Lett. B, 365 (1996), 119-124.  doi: 10.1016/0370-2693(95)01254-0.

[26]

H. B. Nielsen and P. Olesen, Vortex-line models for dual-strings, Nucl. Phys. B, 61 (1973), 45-61. 

[27]

S. Paul and A. Khare, Charged vortices in an abelian Higgs model with Chern-Simons term, Phys. Lett. B, 174 (1986), 420-422.  doi: 10.1016/0370-2693(86)91028-2.

[28]

J. Spruck and Y. Yang, The existence of non-topological solitons in the self-dual Chern-Simons theory, Commun. Math. Phys., 149 (1992), 361-376.  doi: 10.1007/BF02097630.

[29]

G. Tarantello, Selfdual Maxwell-Chern-Simons vortices, Milan J. Math., 72 (2004), 29-80.  doi: 10.1007/s00032-004-0030-9.

[30]

G. Tarantello, Uniqueness of self-dual periodic Chern-Simons vortices of topological-type, Calc. Var. P.D.E., 29 (2007), 191-217.  doi: 10.1007/s00526-006-0062-9.

[31]

G. Tarantello, Selfdual gauge field vortices an analytical approach, In Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser, Boston, 2008.

[32]

M. Torres, Bogomol'nyi limit for nontopological solitons in a Chern-Simons model with anomalous magnetic moment, Phys. Rev. D, 46 (1992), 2295-2298.  doi: 10.1103/PhysRevD.46.R2295.

[33]

Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer Monograph in Mathematics. Springer, New York, 2001.

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