Skip to main content
SpringerLink
Log in

Condensate solutions of the self-dual O(3) Maxwell–Chern–Simons–Higgs equations with symmetric vacua

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We consider a semilinear elliptic system, called the self-dual O(3) Maxwell–Chern–Simons–Higgs equations, on a two-dimensional flat torus arising from the O(3) sigma gauge field model. The system has three important parameters \(\tau \in [0,1]\), \(\kappa >0\) and \(q>0\). We focus on the case of \(\tau =1\) which yields a different structure from that of \(0 \le \tau <1\). Then, we show that the system possesses a solution for a sufficiently small \(\kappa \) and a very large q on a torus. In the proof, we develop a new method in order to use the topological degree theory. We also prove the Chern–Simons limit for our solutions, i.e., the convergence of solutions as \(q \rightarrow \infty \).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arthur, K., Tchrakian, D., Yang, Y.: Topological and nontopological self-dual Chern–Simons solitons in a gauged \(O(3)\) \(\sigma \) model. Phys. Rev. D. 54, 5245–5258 (1996)

    Article  MathSciNet  Google Scholar 

  2. Aubin, T.: Nonlinear Analysis on Manifolds: Monge Ampere Equations, Grundlehren der mathematischen Wissenschaften, vol. 252. Springer, New York (1982)

    Book  Google Scholar 

  3. Bartolucci, D., Lee, Y., Lin, C.-S., Onodera, M.: Asymptotic analysis of solutions to a gauged \(O(3)\) sigma model. Ann. Inst. Henri Poincare Nonlinear Anal. 32, 651–685 (2015)

    Article  MathSciNet  Google Scholar 

  4. Bethuel, F., Brezis, H., Hélein, F.: Asymptotics for the minimization of a Ginzburg–Landau functional. Calc. Var. Partial Differ. Equ. 1, 123–148 (1993)

    Article  MathSciNet  Google Scholar 

  5. Caffarelli, L., Yang, Y.: Vortex condensation in Chern–Simons–Higgs model: an existence theorem. Commun. Math. Phys. 168, 321–336 (1995)

    Article  MathSciNet  Google Scholar 

  6. Chae, D., Kim, N.: Topological multivortex solutions of the self-dual Maxwell–Chern–Simons–Higgs system. J. Differ. Equ. 134, 154–182 (1997)

    Article  MathSciNet  Google Scholar 

  7. Chae, D., Nam, H.-S.: On the condensate multivortex solutions of the self-dual Maxwell–Chern–Simons \(CP(1)\) model. Ann. Henri Poincaré 2, 887–906 (2001)

    Article  MathSciNet  Google Scholar 

  8. Choe, K.: Multivortex solutions in the Chern–Simons gauged \(O(3)\) sigma model on a doubly periodic domain. J. Math. Anal. Appl. 421, 591–624 (2015)

    Article  MathSciNet  Google Scholar 

  9. Choe, K., Han, J.: Existence and properties of radial solutions in the self-dual Chern–Simons \(O(3)\) sigma model. J. Math. Phys. 52, 1–20 (2011). Article No. 082301

  10. Choe, K., Han, J., Lin, C.-S.: Bubbling solutions for the Chern–Simons gauged \(O(3)\) sigma model in \({\mathbb{R}}^{2}\). Discrete Cont. Dyn. Syst. 34, 2703–2728 (2014)

    Article  Google Scholar 

  11. Choe, K., Han, J., Lin, C.-S., Lin, T.-C.: Uniqueness and solution structure of nonlinear equations arising from the Chern–Simons gauged \(O(3)\) sigma models. J. Differ. Equ. 255, 2136–2166 (2013)

    Article  MathSciNet  Google Scholar 

  12. Choe, K., Kim, N.: Blow-up solutions of the self-dual Chern–Simons–Higgs vortex equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 25, 313–338 (2008)

    Article  MathSciNet  Google Scholar 

  13. Choe, K., Nam, H.-S.: Existence and uniqueness of topological multivortex solutions of the self-dual Chern–Simons \(CP(1)\) model. Nonlinear Anal. 66, 2794–2813 (2007)

    Article  MathSciNet  Google Scholar 

  14. Chiacchio, F., Ricciardi, T.: Multiple vortices for a self-dual \(CP(1)\) Maxwell–Chern–Simons model. Nonlinear Differ. Equ. Appl. 13, 563–584 (2007)

    Article  Google Scholar 

  15. Ghosh, P.K., Ghosh, S.K.: Topological and nontopological solitons in a gauged \(O(3)\) sigma model with Chern–Simons term. Phys. Lett. B 366, 199–204 (1996)

    Article  MathSciNet  Google Scholar 

  16. Han, J., Huh, H.: Existence of topological solutions in the Maxwell gauged O(3) Sigma models. J. Math. Anal. Appl. 386, 61–74 (2012)

    Article  MathSciNet  Google Scholar 

  17. Han, J., Lin, C.-S.: Multiplicity for self-dual condensate solutions in the Maxwell–Chern–Simons O(3) sigma model. Commun. PDE 39, 1424–1450 (2014)

    Article  MathSciNet  Google Scholar 

  18. Han, J., Nam, H.-S.: On the topological multivortex solutions of the self-dual Maxwell–Chern–Simons gauged \(O(3)\) sigma model. Lett. Math. Phys. 73, 17–31 (2005)

    Article  MathSciNet  Google Scholar 

  19. Han, J., Song, K.: Existence and asymptotics of topological solutions in the self-dual Maxwell–Chern–Simons \(O(3)\) sigma model. J. Differ. Equ. 250, 204–222 (2011)

    Article  MathSciNet  Google Scholar 

  20. Han, J., Song, K.: Nontopological solutions in the self-dual Maxwell–Chern–Simons gauged \(O(3)\) sigma model. Nonlinear Anal. 118, 22–40 (2015)

    Article  MathSciNet  Google Scholar 

  21. Hong, J., Kim, Y., Pac, P.Y.: Multivortex solutions of the abelian Chern–Simons–Higgs theory. Phys. Rev. Lett. 64, 2230–2233 (1990)

    Article  MathSciNet  Google Scholar 

  22. Jackiw, R., Weinberg, E.J.: Self-dual Chen–Simons vortices. Phys. Rev. Lett. 64, 2234–2237 (1990)

    Article  MathSciNet  Google Scholar 

  23. Kimm, K., Lee, K., Lee, T.: Anyonic Bogomol’nyi solitons in a gauged \(O(3)\) sigma model. Phys. Rev. D 53, 4436–4440 (1996)

    Article  Google Scholar 

  24. Nam, H.-S.: Asymptotics for the condensate multivortex solutions in the self-dual Chern–Simons \(CP(1)\) model. J. Math. Phys. 42, 5698–5712 (2001)

    Article  MathSciNet  Google Scholar 

  25. Nirenberg, L.: Topics in Nonlinear Functional Analysis, Courant Lecture Notes in Mathematics 6. Amer. Math. Soc., Rhode Island (2001)

    Google Scholar 

  26. Nolasco, M., Tarantello, G.: Double vortex condensates in the Chern–Simons–Higgs theory. Calc. Var. PDE 9, 31–94 (1999)

    Article  MathSciNet  Google Scholar 

  27. Ricciardi, T.: Multiplicity for a nonlinear fourth-order elliptic equation in Maxwell–Chern–Simons vortex theory. Differ. Int. Equ. 17, 369–390 (2004)

    MathSciNet  MATH  Google Scholar 

  28. Ricciardi, T., Tarantello, G.: Vortices in the Maxwell–Chern–Simons theory. Commun. Pure Appl. Math. 53, 811–851 (2000)

    Article  MathSciNet  Google Scholar 

  29. Schroers, B.J.: Bogomol’nyi solitons in a gauged \(O(3)\) sigma model. Phys. Lett. B 356, 291–296 (1995)

    Article  MathSciNet  Google Scholar 

  30. Schroers, B.J.: The spectrum of Bogomol’nyi solitons in gauged linear sigma models. Nucl. Phys. B 475, 440–468 (1996)

    Article  Google Scholar 

  31. Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier 15, 189–257 (1965)

    Article  MathSciNet  Google Scholar 

  32. Tarantello, G.: Multiple condensate solutions for the Chern–Simons–Higgs theory. J. Math. Phys. 37, 3769–3796 (1996)

    Article  MathSciNet  Google Scholar 

  33. Yang, S.-G., Chen, Z.-Y., Chern, J.-L.: The solution structure of the \(O(3)\) sigma model in a Maxwell–Chern–Simons theory. J. Math. Phys. 58, Article No. 071503 (2017)

    Article  MathSciNet  Google Scholar 

  34. Yang, Y.: A necessary and sufficient conditions for the existence of multisolitons in a self-dual gauged sigma model. Commun. Math. Phys. 181, 485–506 (1996)

    Article  MathSciNet  Google Scholar 

  35. Yang, Y.: The existence of solitons in gauged sigma models with broken symmetry: some remarks. Lett. Math. Phys. 40, 177–189 (1997)

    Article  MathSciNet  Google Scholar 

  36. Yang, Y.: Solitons in Field Theory and Nonlinear Analysis. Springer Monographs in Mathematics. Springer, New York (2001)

    Book  Google Scholar 

Download references

Acknowledgements

The authors thank the referee for helpful suggestions and comments. The research of Jongmin Han was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2018R1D1A1B07042681), and the research of Kyungwoo Song was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (NRF-2016R1D1A1B03932448).

Author information

Authors and Affiliations

  1. Department of Mathematics, Kyung Hee University, Seoul, 02447, Korea

    Jongmin Han & Kyungwoo Song

Authors
  1. Jongmin Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kyungwoo Song
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kyungwoo Song.

Additional information

Communicated by Y. Giga.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Han, J., Song, K. Condensate solutions of the self-dual O(3) Maxwell–Chern–Simons–Higgs equations with symmetric vacua. Calc. Var. 58, 135 (2019). https://doi.org/10.1007/s00526-019-1564-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-019-1564-6

Mathematics Subject Classification

Search

Navigation