# American Institute of Mathematical Sciences

July  2014, 34(7): 2703-2728. doi: 10.3934/dcds.2014.34.2703

## Bubbling solutions for the Chern-Simons gauged $O(3)$ sigma model in $\mathbb{R}^2$

 1 Department of Mathematics, Inha University, Incheon, 402-751, South Korea 2 Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University, 1 Hoegi-Dong, Dongdaemun-Gu, Seoul, 130-701 3 Taida Institute for Mathematical Sciences(TIMS), National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei, 106, Taiwan

Received  May 2013 Revised  October 2013 Published  December 2013

In this paper, we construct multivortex solutions of the elliptic governing equation for the self-dual Chern-Simons gauged $O(3)$ sigma model in $\mathbb{R}^2$ when the Chern-Simons coupling parameter is sufficiently small, and the location of singular points satisfy suitable conditions. Our solutions show concentration phenomena at some points of the singular points as the coupling parameter tends to zero, and they are locally asymptotically radial near each blow-up point.
Citation: Kwangseok Choe, Jongmin Han, Chang-Shou Lin. Bubbling solutions for the Chern-Simons gauged $O(3)$ sigma model in $\mathbb{R}^2$. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2703-2728. doi: 10.3934/dcds.2014.34.2703
##### References:
 [1] K. Arthur, D. Tchrakian and Y. Yang, Topological and nontopological self-dual Chern-Simons solitons in a gauged $O(3)$ model, Phys. Rev. D (3), 54 (1996), 5245-5258. doi: 10.1103/PhysRevD.54.5245. [2] D. Bartolucci, Y. Lee, C.-S. Lin and M. Onodera, Asymptotic analysis of solutions to a gauged $O(3)$ sigma model, preprint. [3] M. S. Berger and Y. Y. Chen, Symmetric vortices for the Ginzberg-Landau equations of superconductivity and the nonlinear desingularization phenomenon, J. Funct. Anal., 82 (1989), 259-295. doi: 10.1016/0022-1236(89)90071-2. [4] F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices, Progress in Nonlinear Differential Equations and their Applications, 13, Birkhäuser Boston, Inc., Boston, MA, 1994. doi: 10.1007/978-1-4612-0287-5. [5] 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. [6] D. Chae and H.-S. Nam, Multiple existence of the multivortex solutions of the self-dual Chern-Simons $CP(1)$ model on a doubly periodic domain, Lett. Math. Phys., 49 (1999), 297-315. doi: 10.1023/A:1007683108679. [7] 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), 189-221. doi: 10.1007/s00220-002-0691-6. [8] 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), 453-478. doi: 10.1098/rspa.1994.0115. [9] K. Choe, Periodic solutions in the Chern-Simons gauged $O(3)$ sigma model with a symmetric potential, preprint. [10] K. Choe and J. Han, Existence and properties of radial solutions in the self-dual Chern-Simons $O(3)$ sigma model, J. Math. Phys., 52 (2011), 082301, 20 pp. doi: 10.1063/1.3618327. [11] 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. Diff. Eqns., 255 (2013), 2136-2166. doi: 10.1016/j.jde.2013.06.010. [12] K. Choe and H.-S. Nam, Existence and uniqueness of topological multivortex solutions of the self-dual Chern-Simons $CP(1)$ model, Nonlin. Anal., 66 (2007), 2794-2813. doi: 10.1016/j.na.2006.04.008. [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: 10.1103/PhysRevD.53.4436. [14] C.-S. Lin and S. Yan, Bubbling solutions for relativistic abelian Chern-Simons model on a torus, Comm. Math. Phys., 297 (2010), 733-758. [15] H.-S. Nam, Asymptotics for the condensate multivortex solutions in the self-dual Chern-Simons CP(1) model, J. Math. Phys., 42 (2001), 5698-5712. doi: 10.1063/1.1409962. [16] Y. Yang, The existence of solitons in gauged sigma models with broken symmetry: Some remarks, Lett. Math. Phys., 40 (1997), 177-189. doi: 10.1023/A:1007363726173.

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##### References:
 [1] K. Arthur, D. Tchrakian and Y. Yang, Topological and nontopological self-dual Chern-Simons solitons in a gauged $O(3)$ model, Phys. Rev. D (3), 54 (1996), 5245-5258. doi: 10.1103/PhysRevD.54.5245. [2] D. Bartolucci, Y. Lee, C.-S. Lin and M. Onodera, Asymptotic analysis of solutions to a gauged $O(3)$ sigma model, preprint. [3] M. S. Berger and Y. Y. Chen, Symmetric vortices for the Ginzberg-Landau equations of superconductivity and the nonlinear desingularization phenomenon, J. Funct. Anal., 82 (1989), 259-295. doi: 10.1016/0022-1236(89)90071-2. [4] F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices, Progress in Nonlinear Differential Equations and their Applications, 13, Birkhäuser Boston, Inc., Boston, MA, 1994. doi: 10.1007/978-1-4612-0287-5. [5] 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. [6] D. Chae and H.-S. Nam, Multiple existence of the multivortex solutions of the self-dual Chern-Simons $CP(1)$ model on a doubly periodic domain, Lett. Math. Phys., 49 (1999), 297-315. doi: 10.1023/A:1007683108679. [7] 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), 189-221. doi: 10.1007/s00220-002-0691-6. [8] 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), 453-478. doi: 10.1098/rspa.1994.0115. [9] K. Choe, Periodic solutions in the Chern-Simons gauged $O(3)$ sigma model with a symmetric potential, preprint. [10] K. Choe and J. Han, Existence and properties of radial solutions in the self-dual Chern-Simons $O(3)$ sigma model, J. Math. Phys., 52 (2011), 082301, 20 pp. doi: 10.1063/1.3618327. [11] 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. Diff. Eqns., 255 (2013), 2136-2166. doi: 10.1016/j.jde.2013.06.010. [12] K. Choe and H.-S. Nam, Existence and uniqueness of topological multivortex solutions of the self-dual Chern-Simons $CP(1)$ model, Nonlin. Anal., 66 (2007), 2794-2813. doi: 10.1016/j.na.2006.04.008. [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: 10.1103/PhysRevD.53.4436. [14] C.-S. Lin and S. Yan, Bubbling solutions for relativistic abelian Chern-Simons model on a torus, Comm. Math. Phys., 297 (2010), 733-758. [15] H.-S. Nam, Asymptotics for the condensate multivortex solutions in the self-dual Chern-Simons CP(1) model, J. Math. Phys., 42 (2001), 5698-5712. doi: 10.1063/1.1409962. [16] Y. Yang, The existence of solitons in gauged sigma models with broken symmetry: Some remarks, Lett. Math. Phys., 40 (1997), 177-189. doi: 10.1023/A:1007363726173.
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