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

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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

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 & Continuous Dynamical Systems - A, 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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