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http://dx.doi.org/10.4134/BKMS.b150302

REMARKS ON NONTOPOLOGICAL SOLUTIONS IN THE SELF-DUAL CHERN-SIMONS GAUGED O(3) SIGMA MODELS

Choi, Nari (Department of Mathematics Kyung Hee University)
Han, Jongmin (Department of Mathematics Kyung Hee University)

Publication Information

Bulletin of the Korean Mathematical Society / v.53, no.3, 2016 , pp. 765-777 More about this Journal

Abstract

In this paper, we prove the existence of nontopological solutions to the self-dual equations arising from the Chern-Simons gauged O(3) sigma models. The property of solutions depends on a parameter $${\tau}{\in}[-1,1$ $]$ $$$ appearing in the nonlinear term. The case ${\tau}=1$ lies on the borderline for the existence of solutions in the previous results [4, 5, 7]. We prove the existence of solutions in this case when there are only vortex points. Moreover, if $-1{\leq}{\tau}$ <1, we establish solutions which are perturbed from the solutions of singular Liouville equations.

Keywords

Chern-Simons gauged O(3) sigma model; nontopological solutions;

Citations & Related Records

Reference

1	D. Chae and O. Y. Imanuvilov, The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory, Comm. Math. Phys. 215 (2000), no. 1, 119-142. DOI
2	H. Chan, C.-C. Fu, and C.-S. Lin, Non-topological multi-vortex solutions to the self-dual Chern-Simons-Higgs equation, Comm. Math. Phys. 231 (2002), no. 2, 189-221. DOI
3	X. Chen, S. Hastings, J. B. McLeod, and Y. Yang, A nonlinear elliptic equation arising from gauge field theory and cosmology, Proc. Roy. Soc. London Ser. A 446 (1994), no. 1928, 453-478. DOI
4	K. Choe, Existence of nontopological solutions in the Chern-Simons gauged O(3) sigma models, Preprint.
5	K. Choe and J. Han, Existence and properties of radial solutions in the self-dual ChernSimons O(3) sigma model, J. Math. Phys. 52 (2011), no. 8, 082301, 20 pp.
6	K. Choe, J. Han, and C.-S. Lin, Bubbling solutions for the Chern-Simons gauged O(3) sigma model in $\mathbb{R}^2$ , Discrete Contin. Dyn. Syst. 34 (2014), no. 7, 2703-2728. DOI
7	K. Choe, J. Han, C.-S. Lin, and T.-C. Lin, Uniqueness and solution structure of nonlinear equations arising from the Chern-Simons gauged O(3) sigma models, J. Differential Equations 255 (2013), no. 8, 2136-2166. DOI
8	K. Choe, N. Kim, and C.-S. Lin, Existence of self-dual non-topological solutions in the Chern-Simons Higgs model, Ann. Inst. H. Poincare Anal. Non Lineaire 28 (2011), no. 6, 837-852. DOI
9	K. Choe and H.-S. Nam, Existence and uniqueness of topological multivortex solutions of the self-dual Chern-Simons CP(1) model, Nonlinear Anal. 66 (2007), no. 12, 2794-2813. DOI
10	J. Han and H. Huh, Existence of solutions to the self-dual equations in the Maxwell gauged O(3) sigma model, J. Math. Anal. Appl. 386 (2012), no. 1, 61-74. DOI
11	J. Hong, Y. Kim, and P. Y. Pac, Multi-vortex solutions of the Abelian Chern-Simons-Higgs theory, Phys. Rev. Lett. 64 (1990), no. 19, 2230-2233. DOI
12	R. Jackiw and E. J. Weinberg, Self-dual Chern-Simons vortices, Phys. Rev. Lett. 64 (1990), no. 19, 2234-2237. DOI
13	K. Kimm, K. Lee, and T. Lee, Anyonic Bogomol'nyi solitons in a gauged O(3) sigma model, Phys. Rev. D 53 (1996), 4436-4440. DOI
14	W.-M. Ni, On the elliptic equation ${\Delta}u$ + $K(x)u^{(n+1)/(n-2)}$ = 0, its generalizations, and applications in geometry, Indiana Univ. Math. J. 31 (1982), no. 4, 493-529. DOI
15	J. Spruck and Y. Yang, The existence of nontopological solitons in the self-dual ChernSimons theory, Comm. Math. Phys. 149 (1992), no. 2, 361-376. DOI
16	G. Tarantello, Selfdual Gauge Field Vortices, Progress in Nonlinear Differential Equations and their applicatitons Vol 72, Birkhauser, 2008.
17	Y. Yang, The existence of solitons in gauged sigma models with broken symmetry: Some remarks, Lett. Math. Phys. 40 (1997), no. 2, 177-189. DOI