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Energy Splitting, Substantial Inequality, and Minimization for the Faddeev and Skyrme Models

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Abstract

In this paper, we prove that the Faddeev energy E 1 at the unit Hopf charge is attainable. The proof is based on utilizing an important inequality called the substantial inequality in our previous paper which describes how the Faddeev energy splits into its sublevels in terms of energy and topology when compactness fails. With the help of an optimal Sobolev estimate of the Faddeev energy lower bound and an upper bound of E 1, we show that E 1 is attainable. For the two-dimensional Skyrme model, we prove that the substantial inequality is also valid, which allows us to greatly improve the range of the coupling parameters for the existence of unit-charge solitons previously guaranteed in a smaller range of the coupling parameters by the validity of the concentration-compactness method.

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References

  1. Aubin T. (1976) Problemes isoperimetriques et espaces de Sobolev. J. Diff. Geom. 11, 573–598

    MATH  MathSciNet  Google Scholar 

  2. Battye R.A., Sutcliffe P.M. (1998) Knots as stable solutions in a three-dimensional classical field theory. Phys. Rev. Lett. 81, 4798–4801

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. Battye R.A., Sutcliffe P.M. (1999) Solitons, links and knots. Proc. Roy. Soc. A 455, 4305–4331

    Article  MATH  MathSciNet  Google Scholar 

  4. Cho Y.M. (2001) Monopoles and knots in Skyrme theory. Phys. Rev. Ltt. 87: 252001

    Article  ADS  Google Scholar 

  5. Evans L.C.: Weak Convergence Methods for Nonlinear Partial Differential Equations. Regional Conference Series in Math. No. 74, Providence, RI: A. M. S., 1990

  6. Faddeev L.: Einstein and several contemporary tendencies in the theory of elementary particles. In: Relativity, Quanta, and Cosmology, Vol. 1, edited by M. Pantaleo F. de Finis, Newyork: Johnson Reprint Co., 1979, pp. 247–266

  7. Faddeev L.: Knotted solitons. In: Proc. Internat. Congress Mathematicians, Vol. I, Beijing: Higher Ed. Press, 2002, pp. 235–244

  8. Faddeev L., Niemi A.J. (1997) Stable knot-like structures in classical field theory. Nature 387, 58–61

    Article  ADS  Google Scholar 

  9. Jackiw R. (1977) Quantum meaning of classical field theory. Rev. Mod. Phys. 49, 681–706

    Article  MathSciNet  ADS  Google Scholar 

  10. Jackiw R.: Chern-Simons integral as a surface term. http://arxiv.org/list/math-ph/0408051, 2004

  11. Kundu A., Rybakov P. (1982) Closed-vortex-type solitons with Hopf index. J. Phys. A Math. Gen. 15, 269–275

    Article  MathSciNet  ADS  Google Scholar 

  12. Lin F., Yang Y. (2004) Existence of energy minimizers as stable knotted solitons in the Faddeev model. Commun. Math. Phys. 249, 273–303

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Lin F., Yang Y. (2004) Existence of two-dimensional Skyrmions via the concentration-compactness method. Comm. Pure Appl. Math. LVII: 1332–1351

    Article  MathSciNet  Google Scholar 

  14. Lions P.L.: The concentration-compactness principle in the calculus of variations. Part I. Ann. Inst. H. Poincar’e – Anal. non linéaire 1, 109–145 (1984); Part II, ibid 1, 223–283 (1984)

  15. Manton N.S. (1987) Geometry of skyrmions. Commun. Math. Phys. 111, 469–478

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. Rosen G. (1971) Minimum value for c in the Sobolev inequality \(\|\phi^3\|\leq c\|\nabla\phi\|^{3*}\). SIAM J. Appl. Math. 21, 30–32

    Article  MATH  MathSciNet  Google Scholar 

  17. Shabanov S.V. (2002) On a low energy bound in a class of chiral field theories with solitons. J. Math. Phys. 43: 4127–4134

    Article  MATH  MathSciNet  ADS  Google Scholar 

  18. Skyrme T.H.R. (1961) A nonlinear field theory. Proc. Roy. Soc. A 260, 127–138

    MATH  MathSciNet  ADS  Google Scholar 

  19. Skyrme T.H.R. (1961) Particle states of a quantized meson field. Proc. Roy. Soc. A 262, 237–245

    MATH  MathSciNet  ADS  Google Scholar 

  20. Skyrme T.H.R. (1962) A unified field theory of mesons and baryons. Nucl. Phys. 31, 556–569

    Article  MathSciNet  Google Scholar 

  21. Skyrme T.H.R. (1988) The origins of Skyrmions. Internat. J. Mod. Phys. A 3, 2745–2751

    Article  MathSciNet  ADS  Google Scholar 

  22. Tartar L.: Compensated compactness and applications to partial differential equations. In: Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, edited by R.J. Knops, London: Pitman, 1979, pp. 136–212

  23. Talenti G. (1976) Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 352–372

    MathSciNet  Google Scholar 

  24. Vakulenko A.F., Kapitanski L.V. (1979) Stability of solitons in S 2 nonlinear σ-model. Sov. Phys. Dokl. 24, 433–434

    MATH  Google Scholar 

  25. Ward R.S. (1999) Hopf solitons on S 3 and R 3. Nonlinearity 12, 241–246

    Article  MATH  MathSciNet  ADS  Google Scholar 

  26. Whitehead J.H.C. (1947) An expression of Hopf’s invariant as an integral. Proc. Nat. Acad. Sci. 33, 117–123

    Article  MATH  MathSciNet  ADS  Google Scholar 

  27. Zahed I., Brown G.E. (1986) The Skyrme model. Phys. Rep. 142, 1–102

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, New York, New York, 10021, USA

    Fanghua Lin

  2. Department of Mathematics, Polytechnic University, Brooklyn, New York, 11201, USA

    Yisong Yang

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  1. Fanghua Lin
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  2. Yisong Yang
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Correspondence to Yisong Yang.

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Communicated by H.-T. Yau

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Lin, F., Yang, Y. Energy Splitting, Substantial Inequality, and Minimization for the Faddeev and Skyrme Models. Commun. Math. Phys. 269, 137–152 (2007). https://doi.org/10.1007/s00220-006-0123-0

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  • DOI: https://doi.org/10.1007/s00220-006-0123-0

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