American Institute of Mathematical Sciences

Advanced
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 & 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[33]

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

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[33]

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

[1]

Youngae Lee. Non-topological solutions in a generalized Chern-Simons model on torus. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1315-1330. doi: 10.3934/cpaa.2017064
[2]

Hartmut Pecher. Local solutions with infinite energy of the Maxwell-Chern-Simons-Higgs system in Lorenz gauge. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2193-2204. doi: 10.3934/dcds.2016.36.2193
[3]

Youyan Wan, Jinggang Tan. The existence of nontrivial solutions to Chern-Simons-Schrödinger systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2765-2786. doi: 10.3934/dcds.2017119
[4]

Jianjun Yuan. On the well-posedness of Maxwell-Chern-Simons-Higgs system in the Lorenz gauge. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2389-2403. doi: 10.3934/dcds.2014.34.2389
[5]

Kwangseok Choe, Jongmin Han, Chang-Shou Lin. Bubbling solutions for the Chern-Simons gauged $O(3)$ sigma model in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2703-2728. doi: 10.3934/dcds.2014.34.2703
[6]

Hartmut Pecher. The Chern-Simons-Higgs and the Chern-Simons-Dirac equations in Fourier-Lebesgue spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4875-4893. doi: 10.3934/dcds.2019199
[7]

Hyungjin Huh. Towards the Chern-Simons-Higgs equation with finite energy. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1145-1159. doi: 10.3934/dcds.2011.30.1145
[8]

Nikolaos Bournaveas, Timothy Candy, Shuji Machihara. A note on the Chern-Simons-Dirac equations in the Coulomb gauge. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2693-2701. doi: 10.3934/dcds.2014.34.2693
[9]

José Antonio Carrillo, Stefano Lisini, Edoardo Mainini. Uniqueness for Keller-Segel-type chemotaxis models. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1319-1338. doi: 10.3934/dcds.2014.34.1319
[10]

Lucio Boccardo, Alessio Porretta. Uniqueness for elliptic problems with Hölder--type dependence on the solution. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1569-1585. doi: 10.3934/cpaa.2013.12.1569
[11]

Sigmund Selberg, Achenef Tesfahun. Global well-posedness of the Chern-Simons-Higgs equations with finite energy. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2531-2546. doi: 10.3934/dcds.2013.33.2531
[12]

Kwangseok Choe, Hyungjin Huh. Chern-Simons gauged sigma model into $ \mathbb{H}^2 $ and its self-dual equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4613-4646. doi: 10.3934/dcds.2019189
[13]

Jacek Banasiak, Eddy Kimba Phongi. Canard-type solutions in epidemiological models. Conference Publications, 2015, 2015 (special) : 85-93. doi: 10.3934/proc.2015.0085
[14]

Hongbin Chen, Yi Li. Existence, uniqueness, and stability of periodic solutions of an equation of duffing type. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 793-807. doi: 10.3934/dcds.2007.18.793
[15]

Marko Nedeljkov, Sanja Ružičić. On the uniqueness of solution to generalized Chaplygin gas. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4439-4460. doi: 10.3934/dcds.2017190
[16]

Alberto Farina, Miguel Angel Navarro. Some Liouville-type results for stable solutions involving the mean curvature operator: The radial case. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 1233-1256. doi: 10.3934/dcds.2020076
[17]

Tomás Caraballo, Xiaoying Han. A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1079-1101. doi: 10.3934/dcdss.2015.8.1079
[18]

Igor Chueshov, Irena Lasiecka. Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 777-809. doi: 10.3934/dcds.2006.15.777
[19]

Alexander Quaas, Aliang Xia. Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in $\mathbb{R}^N$ involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2653-2668. doi: 10.3934/dcds.2017113
[20]

Vladislav Kruglov, Dmitry Malyshev, Olga Pochinka. Topological classification of $Ω$-stable flows on surfaces by means of effectively distinguishable multigraphs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4305-4327. doi: 10.3934/dcds.2018188

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (77)
  • HTML views (189)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences