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Periodic Maxwell–Chern–Simons vortices with concentrating property

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In order to study electrically and magnetically charged vortices in fractional quantum Hall effect and anyonic superconductivity, the Maxwell–Chern–Simons (MCS) model was introduced by Lee et al. (Phys Lett B 252:79–83, 1990) as a unified system of the classical Abelian–Higgs model (AH) and the Chern–Simons (CS) model. In this article, the first goal is to obtain the uniform (CS) limit result of (MCS) model with respect to the Chern–Simons parameter, without any restriction on either a particular class of solutions or the number of vortex points, as the Chern–Simons mass scale tends to infinity. The most important step for this purpose is to derive the relation between the Higgs field and the neutral scalar field. Our (CS) limit result also provides the critical clue to answer the open problems raised by Ricciardi and Tarantello (Comm Pure Appl Math 53:811–851, 2000) and Tarantello (Milan J Math 72:29–80, 2004), and we succeed to establish the existence of periodic Maxwell–Chern–Simons vortices satisfying the concentrating property of the density of superconductive electron pairs. Furthermore, we expect that the (CS) limit analysis in this paper would help to study the stability, multiplicity, and bubbling phenomena for solutions of the (MCS) model.

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References

  1. Abrikosov, A.A.: On the magnetic properties of superconductors of the second group. Sov. Phys. JETP 5, 1174–1182 (1957)

    Google Scholar 

  2. Bartolucci, D., Chen, C.-C., Lin, C.-S., Tarantello, G.: Profile of blow-up solutions to mean field equations with singular data. Comm. Partial Differ. Equ. 29, 1241–1265 (2004)

    MathSciNet  MATH  Google Scholar 

  3. Bartolucci, D., Tarantello, G.: Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory. Comm. Math. Phys. 229, 3–47 (2002)

    MathSciNet  MATH  Google Scholar 

  4. Bethuel, F., Brezis, H., Helein, F.: Ginzburg–Landau Vortices. Birkhauser, Boston (1994)

    MATH  Google Scholar 

  5. Bogomol’nyi, E.: The stability of classical solutions. Sov. J. Nucl. Phys. 24, 449–454 (1976)

    MathSciNet  Google Scholar 

  6. Boutet de Monvel-Berthier, A., Georgescu, V., Purice, R.: A boundary value problem related to the Ginzburg–Landau model. Comm. Math. Phys. 142, 1–23 (1991)

    MathSciNet  MATH  Google Scholar 

  7. Brezis, H., Merle, F.: Uniform estimates and blow-up behavior for solutions of \(-\Delta u=V(x).e^u\) in two dimensions. Comm. Partial Differ. Equ. 16, 1223–1253 (1991)

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

    MathSciNet  MATH  Google Scholar 

  9. Chae, D., Chae, M.: The global existence in the Cauchy problem of the Maxwell–Chern–Simons–Higgs system. J. Math. Phys. 43, 5470–5482 (2002)

    MathSciNet  MATH  Google Scholar 

  10. Chae, D., Choe, K.: Global existence in the Cauchy problem of the relativistic Chern–Simons–Higgs theory. Nonlinearity 15, 747–758 (2002)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  12. Chae, D., Kim, N.: Vortex condensates in the relativistic self-dual Maxwell–Chern–Simons–Higgs system, RIM-GARC preprint 97-50, Seoul National University

  13. Chae, D., Imanuvilov, Y.: The existence of non-topological multivortex solutions in the relativistic self-dual Chern–Simons theory. Comm. Math. Phys. 215, 119–142 (2000)

    MathSciNet  MATH  Google Scholar 

  14. Chan, H., Fu, C.C., Lin, C.S.: Non-topological multi-vortex solutions to the self-dual Chern–Simons–Higgs equation. Comm. Math. Phys. 231, 189–221 (2002)

    MathSciNet  MATH  Google Scholar 

  15. Chen, W., Li, C.: Qualitative properties of solutions to some nonlinear elliptic equations in \({\mathbb{R}}^2\). Duke Math. J. 71, 427–439 (1993)

    MathSciNet  MATH  Google Scholar 

  16. Chen, X., Hastings, S., McLeod, J.B., Yang, Y.: A nonlinear elliptic equation arising from gauge field theory and cosmology. Proc. Roy. Soc. Lond. A 446, 453–478 (1994)

    MathSciNet  MATH  Google Scholar 

  17. Choe, K.: Existence of multivortex solutions in the self-dual-Higgs theory in a background metric. J. Math. Phys. 42, 5150–5162 (2001)

    MathSciNet  MATH  Google Scholar 

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

  19. Choe, K.: Asymptotic behavior of condensate solutions in the Chern–Simons–Higgs theory. J. Math. Phy. 48, 103501 (2007)

    MathSciNet  MATH  Google Scholar 

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

  21. del Pino, M., Esposito, P., Figueroa, P., Musso, M.: Nontopological condensates for the self-dual Chern–Simons–Higgs model. Comm. Pure Appl. Math. 68(7), 1191–1283 (2015)

    MathSciNet  MATH  Google Scholar 

  22. Ding, W., Jost, J., Li, J., Wang, G.: An analysis of the two-vortex case in the Chern–Simons Higgs model. Calc. Var. Partial Differ. Equ. 7, 87–97 (1998)

    MathSciNet  MATH  Google Scholar 

  23. Ding, W., Jost, J., Li, J., Wang, G.: Multiplicity results for the two-sphere Chern–Simons Higgs model on the two-sphere. Comment. Math. Helv. 74, 118–142 (1999)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  25. Dunne, G.: Self-dual Chern–Simons theories. Lecture Notes in Physics, New series m, Monographs, m36. Springer, New York, (1995)

  26. Fan, Y.W., Lee, Y., Lin, C.S.: Mixed type solutions of the \(SU(3).\) models on a torus. Comm. Math. Phys. 343(1), 233–271 (2016)

  27. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. vol. 224, second ed. Springer, Berlin (1983)

  28. Han, J.: Asymptotics for the vortex condensate solutions in Chern–Simons–Higgs theory. Asymptotic Anal. 28, 31–48 (2001)

    MathSciNet  MATH  Google Scholar 

  29. Han, J.: Asymptotic limit for condensate solutions in the Abelian Chern–Simons Higgs model. Proc. Am. Math. Soc. 131, 1839–1845 (2003)

    MathSciNet  MATH  Google Scholar 

  30. Han, J.: Asymptotic limit for condensate solutions in the Abelian Chern–Simons Higgs model II. Proc. Am. Math. Soc. 131, 3827–3832 (2003)

    MathSciNet  MATH  Google Scholar 

  31. Han, J.: Topological solutions in the self-dual Chern–Simons–Higgs theory in a background metric. Lett. Math. Phys. 65, 37–47 (2003)

    MathSciNet  MATH  Google Scholar 

  32. Han, J., Kim, N.: Nonself-dual Chern–Simons and Maxwell–Chern–Simons vortices on bounded domains. J. Funct. Anal. 221(1), 167–204 (2005)

    MathSciNet  MATH  Google Scholar 

  33. Han, J., Jang, J.: Self-dual Chern–Simons vortices on bounded domains. Lett. Math. Phys. 64, 45–56 (2003)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  36. Jaffe, A., Taubes, C.H.: Vortices and Monopoles. Birkhauser, Boston (1980)

    MATH  Google Scholar 

  37. Kim, S.: Solitons of the self-dual Chern–Simons theory on a cylinder. Lett. Math. Phys. 61, 113–122 (2002)

    MathSciNet  MATH  Google Scholar 

  38. Kim, S., Kim, Y.: Self-dual Chern–Simons vortices on Riemann surfaces. J. Math. Phys. 43, 2355–2362 (2002)

    MathSciNet  MATH  Google Scholar 

  39. Kurata, K.: Existence of nontopological solutions for a nonlinear elliptic equation from Chern–Simons–Higgs theory in a general background metric. Differ. Integral Equ. 14, 925–935 (2001)

    MATH  Google Scholar 

  40. Landau, L., Lifschitz, E.: The Classical Theory of Fields. Addison-Wesley, Cambridge MA (1951)

    Google Scholar 

  41. Lee, C., Lee, K., Min, H.: Self-dual Maxwell–Chern–Simons solitons. Phys. Lett. B 252, 79–83 (1990)

    MathSciNet  Google Scholar 

  42. Lee, Y., Lin, C.S., Yang, W.: Existence of bubbling solutions without mass concentration. Ann. Inst. Fourier (Grenoble) 69(2), 895–940 (2019)

    MathSciNet  MATH  Google Scholar 

  43. Lin, C.S., Yan, S.: Bubbling solutions for relativistic abelian Chern–Simons model on a torus. Comm. Math. Phys. 297, 733–758 (2010)

    MathSciNet  MATH  Google Scholar 

  44. Lin, C.S., Yan, S.: Bubbling solutions for the \(SU(3)\) Chern–Simons Model on a torus. Comm. Pure Appl. Math. 66, 991–1027 (2013)

    MathSciNet  MATH  Google Scholar 

  45. Moon, S., Lee, Y., Huang, H.Y.: Bubbling mixed type solutions of ageneral \({\bf 2 \times 2}\) non-Abelian Chern–Simons–Higgs systemover a torus, preprint

  46. Nielsen, H., Olesen, P.: Vortex-Line models for dual strings. Nucl. Phys. B 61, 45–61 (1973)

    Google Scholar 

  47. Nirenberg, L.: Topics in nonlinear functional analysis. With a chapter by E. Zehnder. Notes by R. A. Artino. Lecture Notes, 1973-1974. Courant Institute of Mathematical Sciences, New York University, New York, (1974)

  48. Nolasco, M., Tarantello, G.: On a sharp type inequality on two dimensional compact manifolds. Arch. Rational Mech. Anal. 145, 161–195 (1998)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  50. Pacard, F., Riviere, T.: Linear and nonlinear aspects of vortices. The Ginzburg–Landau model. Progress in Nonlinear Differential Equations and their Applications 39 Birkhauser Boston, Inc., Boston, MA, (2000)

  51. Ricciardi, T.: Asymptotics for Maxwell–Chern–Simons multivortices. Nonlinear Anal. 50, 1093–1106 (2002)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  53. Schiff, J.: Integrability of Chern–Simons–Higgs and Abelian Higgs vortex equations in a background metric. J. Math. Phys. 32, 753–761 (1991)

    MathSciNet  MATH  Google Scholar 

  54. Spruck, J., Yang, Y.: The existence of nontopological solitons in the self-dual Chern–Simons theory. Comm. Math. Phys. 149, 361–376 (1992)

    MathSciNet  MATH  Google Scholar 

  55. Spruck, J., Yang, Y.: Topological solutions in the self-dual Chern–Simons theory. Ann. Inst. H. Poincaré Anal. Non Lineaire 12, 75–97 (1995)

    MathSciNet  MATH  Google Scholar 

  56. Struwe, M., Tarantello, G.: On multivortex solutions in Chern-Simons gauge theory, Boll. Uni. Mat. Ital. Sez. B Artic. Ric. Mat. (8). 1, 109-121 (1998)

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

    MathSciNet  MATH  Google Scholar 

  58. Tarantello, G.: Selfdual Maxwell–Chern–Simons vortices. Milan J. Math. 72, 29–80 (2004)

    MathSciNet  MATH  Google Scholar 

  59. Taubes, C.H.: Arbitrary N-vortex solutions to the first order Ginzburg–Landau equations. Comm. Math. Phys. 72(3), 277–292 (1980)

    MathSciNet  MATH  Google Scholar 

  60. ’t Hooft, G.: A property of electric and magnetic flux in nonabelian gauge theories. Nucl. Phys. B 153, 141–160 (1979)

  61. Wang, R.: The existence of Chern–Simons vortices. Comm. Math. Phys. 137, 587–597 (1991)

    MathSciNet  MATH  Google Scholar 

  62. Wang, S., Yang, Y.: Abrikosov’s vortices in the critical coupling. SIAM J. Math. Anal. 23, 1125–1140 (1992)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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Acknowledgements

W. Ao was supported by NSFC (Nos. 11801421 and 11631011). O. Kwon was supported by Young Researcher Program through the National Research Foundation of Korea (NRF) (No. NRF-2016R1C1B2014942). Y. Lee was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIT) (No. NRF-2018R1C1B6003403).

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Wuhan University, Wuhan, 430072, People’s Republic of China

    Weiwei Ao

  2. Department of Mathematics, Chungbuk National University, Chungdae-ro 1, Seowon-gu, Cheongju, Chungbuk, 362-763, South Korea

    Ohsang Kwon

  3. Department of Mathematics Education, Teachers College, Kyungpook National University, Daegu, South Korea

    Youngae Lee

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  1. Weiwei Ao
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  2. Ohsang Kwon
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  3. Youngae Lee
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Correspondence to Ohsang Kwon.

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Communicated by Y. Giga.

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Ao, W., Kwon, O. & Lee, Y. Periodic Maxwell–Chern–Simons vortices with concentrating property. Math. Ann. 381, 1885–1942 (2021). https://doi.org/10.1007/s00208-020-02057-7

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