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On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory
a. | Department of Mathematics, National Central University, Chung-Li 32001, Taiwan |
b. | Department of Mathematics, National Changhua University of Education, Changhua 500, Taiwan |
In this paper, by constructing a family of approximation solutions and applying a specific version of the Implicit Function Theorem (please see, e.g. [
References:
[1] |
W. Ao, C.-S. Lin and J. Wei, On non-topological solutions of the A2 and B2 Chern-Simons system, Mem. Amer. Math. Soc., 239 (2016), 1132. Google Scholar |
[2] |
D. Chae and O. Yu. Imanuvilov, The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory, Comm. Math. Phys., 215 (2000), 119–142.
doi: 10.1007/s002200000302. |
[3] |
D. Chae, On the elliptic system arising from a self-gravitating Born-Infeld Abelian Higgs theory, Nonlinearity, 18 (2005), 1823–1833. Google Scholar |
[4] |
W. Chen and C. Li, Qualitative properties of solutions to some nonlinear elliptic equations in $ {\mathbb R}^2$, Duke Math. J., 71 (1993), 427–439.
doi: 10.1215/S0012-7094-93-07117-7. |
[5] |
K.-S. Cheng and C.-S. Lin, On the conformal Gaussian curvature equation in $ {\mathbb R}^2$, J. Diff. Eqns, 146 (1998), 226–250.
doi: 10.1006/jdeq.1998.3424. |
[6] |
J.-L. Chern and S.-G. Yang, Evaluating solutions on an elliptic problem in a gravitational gauge field theory, Journal of Functional Analysis, 265 (2013), 1240-1263. Google Scholar |
[7] |
J.-L. Chern and S.-G. Yang, A survey of solutions in a gravitational Born-Infeld theory, Journal of Mathematical Physics, 55 (2014), 031501, 24pp.
doi: 10.1063/1.4867618. |
[8] |
K. Choe, Uniqueness of the topological multivortex solution in the self-dual in the Chern-Simons Theorem, J. Math. Phys, 46 (2005), 012305. Google Scholar |
[9] |
K. Choe, N. Kim and C.-S. Lin, Existence of self-dual non-topological solutions in the Chern-Simons Higgs model, Ann.Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 28 (2011), 837–852.
doi: 10.1016/j.anihpc.2011.06.003. |
[10] |
K. Choe, N. Kim and C.-S. Lin, Self-dual symmetric nontopological solutions in the $SU(3)$ model in $ {\mathbb R}^2$, Commun. Math. Phys., 33 (2015), 1-37. Google Scholar |
[11] |
K. Choe, N. Kim and C.-S. Lin, Existence of mixed type solutions in the $SU(3)$ Chern-Simons theory in $ {\mathbb R}^2$, Calc. Var. Partial Differential Equations, 56 (2017), Art. 17, 30 pp.
doi: 10.1007/s00526-017-1119-7. |
[12] |
K. Choe, N. Kim, Y. Lin and C.-S. Lin,
Existence of mixed type solutions in the Chern-Simons gauge theory of rank two in $ {\mathbb R}^2$, Journal of Functional Analysis, 273 (2017), 1734-1761.
doi: 10.1016/j.jfa.2017.05.012. |
[13] |
A. Comtet and G. W. Gibbons, Bogomol'nyi bounds for cosmic strings, Nucl. Phys. B, 299 (1988), 719–733.
doi: 10.1016/0550-3213(88)90370-7. |
[14] |
A. V. Fursikov and O. Yu. Imanuvilov, Local exact boundary controllability of the boussinesq equation, SIAM Journal of Control and Optimization, 36 (1998), 391–421. Google Scholar |
[15] |
Z. Hlousek and and D. Spector, Bogomol'nyi explained, Nucl. Phys. B, 397 (1993), 173. Google Scholar |
[16] |
G. Huang and C.-S. Lin,
The existence of non-topological solutions for a skew-symmetric Chern-Simons system, Indiana Univ. Math. J., 65 (2016), 453-491.
doi: 10.1512/iumj.2016.65.5769. |
[17] |
B. Linet, A vortex-line model for a system of cosmic strings in equilibrium, Gen. Relativ. Gravit., 20 (1988), 451–456.
doi: 10.1007/BF00758120. |
[18] |
L. Nirenberg, Topics in Nonlinear Analysis, Courant Lecture Notes in Math., 6, American Mathematical Society, 2001. Google Scholar |
[19] |
D. Tong and K. Wong, Vortices and Impurities, J. High Energy Phys., 1401 (2014), 090. Google Scholar |
[20] |
Y. Yang, Cosmic strings in a product Abelian gauge field theory, Nucl. Phys. B, 885 (2014), 25–33.
doi: 10.1016/j.nuclphysb.2014.05.013. |
[21] |
Y. Yang, Prescribing zeros and poles on a compact Riemann surface for a gravitationally coupled Abelian gauge field theory, Comm. Math. Phys., 249 (2004), 579–609. Google Scholar |
show all references
References:
[1] |
W. Ao, C.-S. Lin and J. Wei, On non-topological solutions of the A2 and B2 Chern-Simons system, Mem. Amer. Math. Soc., 239 (2016), 1132. Google Scholar |
[2] |
D. Chae and O. Yu. Imanuvilov, The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory, Comm. Math. Phys., 215 (2000), 119–142.
doi: 10.1007/s002200000302. |
[3] |
D. Chae, On the elliptic system arising from a self-gravitating Born-Infeld Abelian Higgs theory, Nonlinearity, 18 (2005), 1823–1833. Google Scholar |
[4] |
W. Chen and C. Li, Qualitative properties of solutions to some nonlinear elliptic equations in $ {\mathbb R}^2$, Duke Math. J., 71 (1993), 427–439.
doi: 10.1215/S0012-7094-93-07117-7. |
[5] |
K.-S. Cheng and C.-S. Lin, On the conformal Gaussian curvature equation in $ {\mathbb R}^2$, J. Diff. Eqns, 146 (1998), 226–250.
doi: 10.1006/jdeq.1998.3424. |
[6] |
J.-L. Chern and S.-G. Yang, Evaluating solutions on an elliptic problem in a gravitational gauge field theory, Journal of Functional Analysis, 265 (2013), 1240-1263. Google Scholar |
[7] |
J.-L. Chern and S.-G. Yang, A survey of solutions in a gravitational Born-Infeld theory, Journal of Mathematical Physics, 55 (2014), 031501, 24pp.
doi: 10.1063/1.4867618. |
[8] |
K. Choe, Uniqueness of the topological multivortex solution in the self-dual in the Chern-Simons Theorem, J. Math. Phys, 46 (2005), 012305. Google Scholar |
[9] |
K. Choe, N. Kim and C.-S. Lin, Existence of self-dual non-topological solutions in the Chern-Simons Higgs model, Ann.Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 28 (2011), 837–852.
doi: 10.1016/j.anihpc.2011.06.003. |
[10] |
K. Choe, N. Kim and C.-S. Lin, Self-dual symmetric nontopological solutions in the $SU(3)$ model in $ {\mathbb R}^2$, Commun. Math. Phys., 33 (2015), 1-37. Google Scholar |
[11] |
K. Choe, N. Kim and C.-S. Lin, Existence of mixed type solutions in the $SU(3)$ Chern-Simons theory in $ {\mathbb R}^2$, Calc. Var. Partial Differential Equations, 56 (2017), Art. 17, 30 pp.
doi: 10.1007/s00526-017-1119-7. |
[12] |
K. Choe, N. Kim, Y. Lin and C.-S. Lin,
Existence of mixed type solutions in the Chern-Simons gauge theory of rank two in $ {\mathbb R}^2$, Journal of Functional Analysis, 273 (2017), 1734-1761.
doi: 10.1016/j.jfa.2017.05.012. |
[13] |
A. Comtet and G. W. Gibbons, Bogomol'nyi bounds for cosmic strings, Nucl. Phys. B, 299 (1988), 719–733.
doi: 10.1016/0550-3213(88)90370-7. |
[14] |
A. V. Fursikov and O. Yu. Imanuvilov, Local exact boundary controllability of the boussinesq equation, SIAM Journal of Control and Optimization, 36 (1998), 391–421. Google Scholar |
[15] |
Z. Hlousek and and D. Spector, Bogomol'nyi explained, Nucl. Phys. B, 397 (1993), 173. Google Scholar |
[16] |
G. Huang and C.-S. Lin,
The existence of non-topological solutions for a skew-symmetric Chern-Simons system, Indiana Univ. Math. J., 65 (2016), 453-491.
doi: 10.1512/iumj.2016.65.5769. |
[17] |
B. Linet, A vortex-line model for a system of cosmic strings in equilibrium, Gen. Relativ. Gravit., 20 (1988), 451–456.
doi: 10.1007/BF00758120. |
[18] |
L. Nirenberg, Topics in Nonlinear Analysis, Courant Lecture Notes in Math., 6, American Mathematical Society, 2001. Google Scholar |
[19] |
D. Tong and K. Wong, Vortices and Impurities, J. High Energy Phys., 1401 (2014), 090. Google Scholar |
[20] |
Y. Yang, Cosmic strings in a product Abelian gauge field theory, Nucl. Phys. B, 885 (2014), 25–33.
doi: 10.1016/j.nuclphysb.2014.05.013. |
[21] |
Y. Yang, Prescribing zeros and poles on a compact Riemann surface for a gravitationally coupled Abelian gauge field theory, Comm. Math. Phys., 249 (2004), 579–609. Google Scholar |
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