Skip to main content
SpringerLink
Log in

Mixed Type Solutions of the \({SU(3)}\) Models on a Torus

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

In this paper, we study mixed-type solutions of \({SU(3)}\) Chern–Simons system (see (1.4) below) on a two dimensional flat torus. Nolasco and Tarantello (Commun Math Phys 213:599–639, 2000), among other things, Nolasco and Tarantello obtained solutions of (1.4) as minimizers of several functionals closely related to (1.4), and showed that if \({N_1+N_2=1}\), then one of those minimizers turns out to be a mixed-type solution, that is, one component tends to \({\ln\frac{1}{2}}\) pointwise a.e. and the other component converges to a solution of a mean field equation. We call these kinds of solutions mixed-type (I) solutions. In this paper, we prove two main results: (i) the asymptotic analysis of mixed-type (I) solutions with arbitrary configuration of vortex points, and (ii) the existence of mixed-type (I) solutions under a non-degenerate condition. This non-degenerate condition also ensures some uniqueness result. In particular, our results imply that when \({N_1+N_2=1}\), there are only two mixed-type (I) solutions of (1.4).

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. Ao, W., Lin, C.S., Wei, J.: On non-topological solutions of the \({A_{2}}\) and \({B_{2}}\) Chern–Simons system. Memoirs Am. Math. Soc. (accepted for publication)

  2. Bezryadina A., Eugenieva E., Chen Z.: Self-trapping and flipping of double-charged vortices in optically induced photonic lattices. Opt. Lett. 31, 2456–2458 (2006)

    Article  ADS  Google Scholar 

  3. Chai C.L., Lin C.S., Wang C.L.: Mean field equations, hyperelliptic curves and modular forms: I. Cambr. J. Math. 3(1-2), 127–274 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Chai, C.L., Lin, C.S., Wang, C.L.: Mean field equations with multiple singularity and algebraic geometry (in preparation)

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Chen C.C., Lin C.S.: Mean field equation of Liouville type with singular data: topological degree. Commun. Pure Appl. Math. 68, 887–947 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Choe K., Kim N., Lin C.S.: Existence of self-dual non-topological solutions in the Chren-Simons-Higgs model. Ann. Inst. H. P. 28, 837–852 (2011)

    ADS  MathSciNet  MATH  Google Scholar 

  11. Choe K., Kim N., Lin C.S.: Self-dual symmetric nontopological solutions in the \({SU(3)}\) model in \({\mathbb{R}^2}\). Commun. Math. Phys. 334, 1–37 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Choe, K., Kim, N., Lin, C.S.: New type of nontopological bubbling solutions in the \({SU(3)}\) Chern–Simons model in \({\mathbb{R}^2}\) (preprint)

  13. Choe, K., Kim, N., Lin, C.S.: Existence of solutions of mixed type in the \({SU(3)}\) Chern–Simons theory in \({\mathbb{R}^2}\) (preprint)

  14. Dunne G.: Mass degeneracies in self-dual models. Phys. Lett. B 345, 452–457 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  15. Dunne G.: Self-dual Chern–Simons Theories. Lecture Notes in Physics. vol. m36. Spring, Berlin (1995)

    Google Scholar 

  16. Dunne G.: Vacuum mass spectra for \({SU(N)}\) self-dual Chern–Simons–Higgs. Nucl. Phys. B 433, 333–348 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order, vol. 224, 2nd ed. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  18. Han J.: Existence of topological multivortex solutions in the self-dual gauge theories. Proc. R. Soc. Edinburgh Sect. A 130, 1293–1309 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Huang H.Y., Lin C.S.: On the Entire radial solutions of the Chern–Simons SU(3) system. Commun. Math. Phys. 327, 815–848 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Inouye S., Gupta S., Rosenband T., Chikkatur A.P., Grlitz A., Gustavson T.L., Leanhardt A.E., Pritchard D.E., Ketterle W.: Observation of vortex phase singularities in Bose–Einstein condensates. Phys. Rev. Lett. 87, 080402 (2001)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Kao H., Lee K.: Self-dual \({SU(3)}\) Chern–Simons higgs systems. Phys. Rev. D 50, 6626–6632 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  24. Kawaguchi Y., Ohmi T.: Splitting instability of a multiply charged vortex in a Bose–Einstein condensate. Phys. Rev. A 70, 043610 (2004)

    Article  ADS  Google Scholar 

  25. Khomskii D.I., Freimuth A.: Charged vortices in high temperature superconductors. Phys. Rev. Lett. 75, 1384–1386 (1995)

    Article  ADS  Google Scholar 

  26. Lang S.: Complex analysis, vol. 103, 4th edn. Springer, Berlin (1999)

    Book  Google Scholar 

  27. Lin C.S., Wang C.L.: Elliptic functions, Green functions and the mean field equations on tori. Ann. Math. (2) 172, 911–954 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lin, C.S., Wang, C.L.: On the minimality of extra critical points of Green functions on flat tori (preprint)

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Lin C.S., Yan S.: Existence of Bubbling solutions for Chern–Simons model on a torus. Arch. Rat. Mech. Anal. 207, 353–392 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lin C.S., Yang Y.: Non-Abelian multiple vortices in supersymmetric field theory. Commun. Math. Phys. 304, 433–457 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Lin C.S., Yang Y.: Sharp existence and uniqueness theorems for non-Abelian multiple vortex solutions. Nucl. Phys. B 846, 650–676 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Matsuda Y., Nozakib K., Kumagaib K.: Charged vortices in high temperature superconductors probed by nuclear magnetic resonance. J. Phys. Chem. Solids 63, 1061–1063 (2002)

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Nolasco M., Tarantello G.: Vortex condensates for the \({SU(3)}\) Chern–Simons theory. Commun. Math. Phys. 213, 599–639 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Shevchenko S.I.: Charged vortices in superfluid systems with pairing of spatially separated carriers. Phys. Rev. B 67, 214515 (2003)

    Article  ADS  Google Scholar 

  37. Sokoloff J.B.: Charged vortex excitations in quantum Hall systems. Phys. Rev. B 31, 1924–1928 (1985)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. ’t Hooft G.: A property of electric and magnetic flux in nonabelian gauge theories. Nucl. Phys. B153, 141–160 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  40. Tarantello G.: Uniqueness of self-dual periodic Chern–Simons vortices of topological-type. Calc. Var. P.D.E. 28, 191–217 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  41. Tarantello G.: Selfdual Gauge Field Vortices. An Analytical Approach. Progress in Nonlinear Differential Equations and their Applications. Birkhauser Boston, Inc., Boston (2008)

    Google Scholar 

  42. Yang Y.: The relativistic non-Abelian Chern–Simons equations. Commun. Math. Phys. 186, 199–218 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Yang Y.: Solitions in Field Theory and Nonlinear Analysis. Springer Monographs in Mathemayics. Springer, New York (2001)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Taida Institute for Mathematical Sciences, National Taiwan University, No.1, Sec. 4, Roosevelt Road, Taipei, 106, Taiwan

    Yen-Wen Fan

  2. Center for Advanced Study in Theoretical Science, National Taiwan University, No.1, Sec. 4, Roosevelt Road, Taipei, 106, Taiwan

    Youngae Lee

  3. Taida Institute for Mathematical Sciences, Center for Advanced Study in Theoretical Science, National Taiwan University, No.1, Sec. 4, Roosevelt Road, Taipei, 106, Taiwan

    Chang-Shou Lin

Authors
  1. Yen-Wen Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Youngae Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chang-Shou Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Youngae Lee.

Additional information

Communicated by H.-T. Yau

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fan, YW., Lee, Y. & Lin, CS. Mixed Type Solutions of the \({SU(3)}\) Models on a Torus. Commun. Math. Phys. 343, 233–271 (2016). https://doi.org/10.1007/s00220-015-2532-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-015-2532-4

Keywords

Search

Navigation