December  2016, 9(6): 2201-2211. doi: 10.3934/dcdss.2016092

Analytical solutions of Skyrme model

1. 

Institute of Contemporary Mathematics, College of Mathematics and Statistics, Henan University, Kaifeng 475001, China

2. 

College of Mathematics and Statistics, Henan University, Kaifeng 475004, China

3. 

Hualuogeng Class, School of Mathematics and System Sciences, Beihang University, Beijing 100191, China

Received  July 2015 Revised  September 2016 Published  November 2016

Exact analytic solutions of the kink soliton equation obtained in a recent interesting study of the classical Skyrme model defined on a simple spherically symmetric background are presented. By a variational method, the existence of spherically symmetric monopole solutions are proved. In particular, all finite-energy kink solitons must be Bogomool'nyi--Prasad--Sommerfield are showed. Moreover, together with numerical analysis, we can clearly see the validity of our theoretical results.
Citation: Ruifeng Zhang, Nan Liu, Man An. Analytical solutions of Skyrme model. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2201-2211. doi: 10.3934/dcdss.2016092
References:
[1]

G. S. Adkins, C. R. Nappi and E. Witten, Static properties of nucleons in the Skyrme model,, Selected Papers, 3 (1983), 317. doi: 10.1142/9789812795922_0020. Google Scholar

[2]

A. P. Balachandran, H. Gomm and R. D. Sorkin, Quantum symmetries from quantum phases, fermions from bosons, $A$ $Z_2$ anomally and Galilean invariance,, Nucl. Phys. B, 281 (1987), 573. doi: 10.1016/0550-3213(87)90420-2. Google Scholar

[3]

E. B. Bogomol'nyi, The stability of calssical solution,, Sov. J. Nucl. Phys., 24 (1976), 449. Google Scholar

[4]

F. Canfora, Nonlinear superposition law and Skyrme crystals,, Phys. Rev. D, 88 (2013). doi: 10.1103/PhysRevD.88.065028. Google Scholar

[5]

F. Canfora and P. Salgado-Rebolledo, Generalized hedgehog ansatz and Gribov copies in regions with nontrivial topologies,, Phys. Rev. D, 87 (2013). doi: 10.1103/PhysRevD.87.045023. Google Scholar

[6]

F. Canfora and H. Maeda, Hedgehog ansatz and its generalization for self-gravitating Skyrmions,, Phys. Rev. D, 87 (2013). doi: 10.1103/PhysRevD.87.084049. Google Scholar

[7]

F. Canfora, F. Correa and J. Zanelli, Exact multi-soliton solutions in the four dimensional Skyrme model,, Phys. Rev. D, 90 (2014). Google Scholar

[8]

S. Chen, Y. Li and Y. Yang, Exact kink solitons in a monopole confinement problem,, Phys, 86 (2012). doi: 10.1103/PhysRevD.86.085030. Google Scholar

[9]

S. Chen, Y. Li and Y. Yang, Exact kink solitons in Skyrme crystals,, Phys, 89 (2014). doi: 10.1103/PhysRevD.89.025007. Google Scholar

[10]

Y. M. Cho, Monopoles and knots in Skyrme theory,, Phys. Rev. Lett., 87 (2001). doi: 10.1103/PhysRevLett.87.252001. Google Scholar

[11]

J. Fukuda and S. Zumer, Quasi-two-dimensional Skyrmion lattices in a chiral nematic liquid crystal,, Nature Communications, 2 (2011). doi: 10.1038/ncomms1250. Google Scholar

[12]

U. Al Khawaja and H. Stoof, Skyrmions in a ferromagnetic Bose-Einstein condensate,, Nature, 411 (2001), 918. doi: 10.1038/35082010. Google Scholar

[13]

E. H. Lieb, Remarks on the Skyrme model,, Proc. Symposia Pure Math., 54 (1993), 379. Google Scholar

[14]

F. Lin and Y. Yang, Existence of energy minimizers as stable knotted solitons in the Faddeev model,, Commun. Math. Phys., 249 (2004), 273. doi: 10.1007/s00220-004-1110-y. Google Scholar

[15]

N. S. Manton, A remark on the scattering of BPS monopoles,, Phys. Lett. B, 110 (1982), 54. doi: 10.1016/0370-2693(82)90950-9. Google Scholar

[16]

N. S. Manton and P. J. Ruback, Skyrmions in flat space and curved space,, Phys. Lett. B, 181 (1986), 137. doi: 10.1016/0370-2693(86)91271-2. Google Scholar

[17]

N. S. Manton, Geometry of Skyrmions,, Commun. Math. Phys., 111 (1987), 469. doi: 10.1007/BF01238909. Google Scholar

[18]

S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii and P. Böni, Skyrmion lattice in a chiral magnet,, Science, 323 (2009), 915. Google Scholar

[19]

H. Pais and J. R. Stone, Exploring the nuclear pasta phase in core-collapse supernova matter,, Phys. Rev. Lett., 109 (2012). doi: 10.1103/PhysRevLett.109.151101. Google Scholar

[20]

M. K. Prasad and C. M. Sommerfield, Exact Classical Solution for the 't Hooft Monopole and the Julia-Zee Dyon,, Phys, 35 (1975), 760. doi: 10.1103/PhysRevLett.35.760. Google Scholar

[21]

T. Skyrme, A non-linear field theory,, Proc. R. Soc. London A, 260 (1961), 127. doi: 10.1098/rspa.1961.0018. Google Scholar

[22]

E. Witten, Global aspects of current algebra,, Nucl. Phys. B, 223 (1983), 422. doi: 10.1016/0550-3213(83)90063-9. Google Scholar

[23]

Y. Yang, Existence of solutions for a generalized Yang-Mills theory,, Lett. Math. Phys., 19 (1990), 257. doi: 10.1007/BF01039320. Google Scholar

[24]

R. Zhang and J. Zhao, On the existence of Skyrme gauge field monopoles,, Nonlinear Anal., 75 (2012), 1679. doi: 10.1016/j.na.2011.04.062. Google Scholar

show all references

References:
[1]

G. S. Adkins, C. R. Nappi and E. Witten, Static properties of nucleons in the Skyrme model,, Selected Papers, 3 (1983), 317. doi: 10.1142/9789812795922_0020. Google Scholar

[2]

A. P. Balachandran, H. Gomm and R. D. Sorkin, Quantum symmetries from quantum phases, fermions from bosons, $A$ $Z_2$ anomally and Galilean invariance,, Nucl. Phys. B, 281 (1987), 573. doi: 10.1016/0550-3213(87)90420-2. Google Scholar

[3]

E. B. Bogomol'nyi, The stability of calssical solution,, Sov. J. Nucl. Phys., 24 (1976), 449. Google Scholar

[4]

F. Canfora, Nonlinear superposition law and Skyrme crystals,, Phys. Rev. D, 88 (2013). doi: 10.1103/PhysRevD.88.065028. Google Scholar

[5]

F. Canfora and P. Salgado-Rebolledo, Generalized hedgehog ansatz and Gribov copies in regions with nontrivial topologies,, Phys. Rev. D, 87 (2013). doi: 10.1103/PhysRevD.87.045023. Google Scholar

[6]

F. Canfora and H. Maeda, Hedgehog ansatz and its generalization for self-gravitating Skyrmions,, Phys. Rev. D, 87 (2013). doi: 10.1103/PhysRevD.87.084049. Google Scholar

[7]

F. Canfora, F. Correa and J. Zanelli, Exact multi-soliton solutions in the four dimensional Skyrme model,, Phys. Rev. D, 90 (2014). Google Scholar

[8]

S. Chen, Y. Li and Y. Yang, Exact kink solitons in a monopole confinement problem,, Phys, 86 (2012). doi: 10.1103/PhysRevD.86.085030. Google Scholar

[9]

S. Chen, Y. Li and Y. Yang, Exact kink solitons in Skyrme crystals,, Phys, 89 (2014). doi: 10.1103/PhysRevD.89.025007. Google Scholar

[10]

Y. M. Cho, Monopoles and knots in Skyrme theory,, Phys. Rev. Lett., 87 (2001). doi: 10.1103/PhysRevLett.87.252001. Google Scholar

[11]

J. Fukuda and S. Zumer, Quasi-two-dimensional Skyrmion lattices in a chiral nematic liquid crystal,, Nature Communications, 2 (2011). doi: 10.1038/ncomms1250. Google Scholar

[12]

U. Al Khawaja and H. Stoof, Skyrmions in a ferromagnetic Bose-Einstein condensate,, Nature, 411 (2001), 918. doi: 10.1038/35082010. Google Scholar

[13]

E. H. Lieb, Remarks on the Skyrme model,, Proc. Symposia Pure Math., 54 (1993), 379. Google Scholar

[14]

F. Lin and Y. Yang, Existence of energy minimizers as stable knotted solitons in the Faddeev model,, Commun. Math. Phys., 249 (2004), 273. doi: 10.1007/s00220-004-1110-y. Google Scholar

[15]

N. S. Manton, A remark on the scattering of BPS monopoles,, Phys. Lett. B, 110 (1982), 54. doi: 10.1016/0370-2693(82)90950-9. Google Scholar

[16]

N. S. Manton and P. J. Ruback, Skyrmions in flat space and curved space,, Phys. Lett. B, 181 (1986), 137. doi: 10.1016/0370-2693(86)91271-2. Google Scholar

[17]

N. S. Manton, Geometry of Skyrmions,, Commun. Math. Phys., 111 (1987), 469. doi: 10.1007/BF01238909. Google Scholar

[18]

S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii and P. Böni, Skyrmion lattice in a chiral magnet,, Science, 323 (2009), 915. Google Scholar

[19]

H. Pais and J. R. Stone, Exploring the nuclear pasta phase in core-collapse supernova matter,, Phys. Rev. Lett., 109 (2012). doi: 10.1103/PhysRevLett.109.151101. Google Scholar

[20]

M. K. Prasad and C. M. Sommerfield, Exact Classical Solution for the 't Hooft Monopole and the Julia-Zee Dyon,, Phys, 35 (1975), 760. doi: 10.1103/PhysRevLett.35.760. Google Scholar

[21]

T. Skyrme, A non-linear field theory,, Proc. R. Soc. London A, 260 (1961), 127. doi: 10.1098/rspa.1961.0018. Google Scholar

[22]

E. Witten, Global aspects of current algebra,, Nucl. Phys. B, 223 (1983), 422. doi: 10.1016/0550-3213(83)90063-9. Google Scholar

[23]

Y. Yang, Existence of solutions for a generalized Yang-Mills theory,, Lett. Math. Phys., 19 (1990), 257. doi: 10.1007/BF01039320. Google Scholar

[24]

R. Zhang and J. Zhao, On the existence of Skyrme gauge field monopoles,, Nonlinear Anal., 75 (2012), 1679. doi: 10.1016/j.na.2011.04.062. Google Scholar

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