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Topological solutions in the self-dual Chern-Simons theory: Existence and approximation. (English) Zbl 0836.35007
Summary: In this paper a globally convergent computational scheme is established to approximate a topological multivortex solution in the recently discovered self-dual Chern-Simons theory in \(\mathbb{R}^2\). Our method which is constructive and numerically efficient finds the most superconducting solution in the sense that its Higgs field has the largest possible magnitude. The method consists of two steps: first one obtains by a convergent monotone iterative algorithm a suitable solution of the bounded domain equations and then one takes the large domain limit and approximates the full plane solutions. It is shown that with a special choice of the initial guess function, the approximation sequence approaches exponentially fast a solution in \(\mathbb{R}^2\). The convergence rate implies that the truncation errors away from local regions are insignificant.

35A35 Theoretical approximation in context of PDEs
35Q40 PDEs in connection with quantum mechanics
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[1] Bogomol’nyi, E., The stability of classical solutions, Sov. J. Nucl. Phys., Vol. 24, 449-454, (1976)
[2] de Vega, H. J.; Schaposnik, F., Electrically charged vortices in nonabelian gauge theories with Chern-Simons term, Phys. Rev. Lett., Vol. 56, 2564-2566, (1986)
[3] Fröhlich, J.; Marchetti, J., Quantum field theory of anyons, Lett. Math. Phys., Vol. 16, 347-358, (1988) · Zbl 0695.58031
[4] Fröhlich, J.; Marchetti, P., Quantum field theory of vortices and anyons, Commun. Math. Phys., Vol. 121, 177-223, (1989) · Zbl 0819.58045
[5] Gilbarg, D.; Trudinger, N., Elliptic partial differential equations of second order, (1977), Springer Berlin · Zbl 0361.35003
[6] Hong, J.; Kim, Y.; Pac, P., Multivortex solutions of the abelian Chern-Simons-Higgs theory, Phys. Rev. Lett., Vol. 64, 2230-2233, (1990) · Zbl 1014.58500
[7] R. Jackiw, Solitons in Chern-Simons/anyons systems, Preprint. · Zbl 0947.81531
[8] Jackiw, R.; Lee, K.; Weinberg, E., Self-dual Chern-Simons solitons, Phys. Rev. D., Vol. 42, 3488-3499, (1990)
[9] R. Jackiw, S.-Y. Pi and E. Weinberg, Topological and non-topological solitons in relativistic and non-relativistic Chern-Simons theory, Preprint.
[10] Jackiw, R.; Weinberg, E., Self-dual Chern-Simons vortices, Phys. Rev. Lett., Vol. 64, 2234-2237, (1990) · Zbl 1050.81595
[11] A. Jaffe and C. H. Taubes, Vortices and Monopoles, Birkhaüser, Boston, 1980. · Zbl 0457.53034
[12] Julia, B.; Zee, A., Poles with both magnetic and electric charges in nonabelian gauge theory, Phys. Rev. D., Vol. 11, 2227-2232, (1975)
[13] Nielsen, H.; Olesen, P., Vortex-iine models for dual-strings, Nucl. Phys. B, Vol. 61, 45-61, (1973)
[14] Paul, S.; Khare, A., Charged vortices in an abelian Higgs model with Chern-Simons term, Phys. Lett. B, Vol. 174, 420-422, (1986)
[15] Paul, S.; Khare, A., Charged vortex of finite energy in nonabelian gauge theories with Chern-Simons term, Phys. Lett. B, Vol. 178, 395-399, (1986)
[16] Spruck, J.; Yang, Y., The existence of non-topological solitons in the self-dual Chern-Simons theory, Commun. Math. Phys., Vol. 149, 361-376, (1992) · Zbl 0760.53063
[17] Taubes, C., On the equivalence of the first and second order equations for gauge theories, Commun. Math. Phys., Vol. 75, 207-227, (1980) · Zbl 0448.58029
[18] Wang, R., The existence of Chern-Simons vórtices, Commun. Math. Phys., Vol. 137, 587-597, (1991) · Zbl 0733.58009
[19] Wang, S.; Yang, Y., Abrikosov’s vortices in the criticai coupling, SIAM J. Math. Anal., Vol. 23, 1125-1140, (1992) · Zbl 0753.35111
[20] Wang, S.; Yang, Y., Solutions of the generalized bogomol’nyi equations via monotone iterations, J. Math. Phys., Vol. 33, 4239-4249, (1992) · Zbl 0767.35085
[21] Yang, Y., Existence, regularity, and asymptotic behavior of the solutions to the Ginzburg-Landau equations on R^3, Commun. Math. Phys., Vol. 123, 147-161, (1989) · Zbl 0678.35086
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