October  2022, 42(10): 4887-4903. doi: 10.3934/dcds.2022077

On the asymptotic behavior of solutions for the self-dual Maxwell-Chern-Simons $ O(3) $ Sigma model

Department of Mathematics, National Changhua University of Education, Changhua 500, Taiwan

*Corresponding author: Zhi-You Chen

Received  April 2021 Revised  November 2021 Published  October 2022 Early access  June 2022

Fund Project: The first author is supported by the Ministry of Science and Technology of Taiwan under the grant MOST 110-2115-M-018-004-MY3

In this paper, we consider the nonlinear equations arising from the self-dual Maxwell-Chern-Simons gauged $ O(3) $ sigma model on (2+1)-dimensional Minkowski space $ {\bf R^{2,1}} $ with the metric $ {\mathrm {diag}}(1,-1,-1) $. We establish the asymptotic behavior of multivortex solutions corresponding to their flux and find the range of the flux for non-topological solutions. Moreover, we prove the radial symmetry property under certain conditions in one vortex point case.

Citation: Zhi-You Chen, Chung-Yang Wang, Yu-Jen Huang. On the asymptotic behavior of solutions for the self-dual Maxwell-Chern-Simons $ O(3) $ Sigma model. Discrete and Continuous Dynamical Systems, 2022, 42 (10) : 4887-4903. doi: 10.3934/dcds.2022077
References:
[1]

A.-A. Bogomol'nyi, The stability of classical solutions, Sov. J. Nucl. Phys., 24 (1976), 449-454. 

[2]

D. Chae and O. Yu. Imanuvilov, Non-topological multivortex solutions to the self-dual Maxwell-Chern-Simons-Higgs systems, J. Funct. Anal., 196 (2002), 87-118.  doi: 10.1006/jfan.2002.3988.

[3]

D. Chae and N. Kim, Vortex condensates in the relativistic self-dual Maxwell-Chern-Simons-Higgs system, RIM-GARC Preprint, (1997), 97–50.

[4]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.

[5]

Z.-Y. Chen and J.-L. Chern, Sharp range of flux and the structure of solutions for the self-dual Maxwell-Chern-Simons $O(3)$ Sigma model, submitted.

[6]

Z.-Y. Chen and J.-L. Chern, The analysis of solutions for Maxwell-Chern-Simons $O(3)$ Sigma model, Calculus of Variations and Partial Differential Equations, 58 (2019), 147.  doi: 10.1007/s00526-019-1590-4.

[7]

K.-S. Cheng and C.-S. Lin, On the asymptotic behavior of solutions of the conformal Gaussian curvature equations in ${\bf{R}}^2$, Math. Annalen, 308 (1997), 119-139.  doi: 10.1007/s002080050068.

[8]

J.-L. ChernZ.-Y. Chen and H.-Y. Shen, Classification of solutions for self-dual Chern-Simons $CP(1)$ Model, J. Math. Phys., 62 (2021), 031510.  doi: 10.1063/5.0022001.

[9]

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.

[10]

K. ChoeJ. HanC.-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), 2136-2166.  doi: 10.1016/j.jde.2013.06.010.

[11]

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), 61-74.  doi: 10.1016/j.jmaa.2011.07.046.

[12]

J. Han and J. Jang, Nontopological bare solutions in the relativistic self-dual Maxwell-Chern-Simons-Higgs model, J. Math. Phys., 46 (2005), 042310, 16 pp. doi: 10.1063/1.1861277.

[13]

J. Han and H.-S. Nam, On the topological multivortex solutions of the self-dual Maxwell-Chern-Simons gauged O(3) sigma model, Lett. Math. Phys., 73 (2005), 17-31.  doi: 10.1007/s11005-005-8443-0.

[14]

J. Han and K. Song, Existence and asymptotic of topological solutions in the self-dual Maxwell-Chern-Simons $O(3)$ Sigma model, J. Diff. Eqn., 250 (2011), 204-222.  doi: 10.1016/j.jde.2010.08.003.

[15]

K. Kim, 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.

[16]

T. Ricciardi, Asymptotics for Maxwell-Chern-Simons multivortices, Nonlinear. Anal. TMA, 50 (2002), 1093-1106.  doi: 10.1016/S0362-546X(01)00752-0.

[17]

T. Ricciardi and G. Tarantello, Vortices in the Maxwell-Chern-Simons theory, Comm. Pure Appl. Math., 53 (2000), 811-851.  doi: 10.1002/(SICI)1097-0312(200007)53:7<811::AID-CPA2>3.0.CO;2-F.

[18]

B.-J. Schroers, Bogomol'nyi solitons in a gauged $O(3)$ sigma model, Phys. Lett. B, 356 (1995), 291-296.  doi: 10.1016/0370-2693(95)00833-7.

[19]

K. Song, Radial symmetry of topological solitons in the self-dual Maxwell-Chern-Simons $O(3)$ Sigma model, Bull. Korean. Math. Soc., 48 (2011), 1111-1117.  doi: 10.4134/BKMS.2011.48.5.1111.

[20]

Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer Monographs in Mathematics, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-6548-9.

show all references

References:
[1]

A.-A. Bogomol'nyi, The stability of classical solutions, Sov. J. Nucl. Phys., 24 (1976), 449-454. 

[2]

D. Chae and O. Yu. Imanuvilov, Non-topological multivortex solutions to the self-dual Maxwell-Chern-Simons-Higgs systems, J. Funct. Anal., 196 (2002), 87-118.  doi: 10.1006/jfan.2002.3988.

[3]

D. Chae and N. Kim, Vortex condensates in the relativistic self-dual Maxwell-Chern-Simons-Higgs system, RIM-GARC Preprint, (1997), 97–50.

[4]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.

[5]

Z.-Y. Chen and J.-L. Chern, Sharp range of flux and the structure of solutions for the self-dual Maxwell-Chern-Simons $O(3)$ Sigma model, submitted.

[6]

Z.-Y. Chen and J.-L. Chern, The analysis of solutions for Maxwell-Chern-Simons $O(3)$ Sigma model, Calculus of Variations and Partial Differential Equations, 58 (2019), 147.  doi: 10.1007/s00526-019-1590-4.

[7]

K.-S. Cheng and C.-S. Lin, On the asymptotic behavior of solutions of the conformal Gaussian curvature equations in ${\bf{R}}^2$, Math. Annalen, 308 (1997), 119-139.  doi: 10.1007/s002080050068.

[8]

J.-L. ChernZ.-Y. Chen and H.-Y. Shen, Classification of solutions for self-dual Chern-Simons $CP(1)$ Model, J. Math. Phys., 62 (2021), 031510.  doi: 10.1063/5.0022001.

[9]

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.

[10]

K. ChoeJ. HanC.-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), 2136-2166.  doi: 10.1016/j.jde.2013.06.010.

[11]

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), 61-74.  doi: 10.1016/j.jmaa.2011.07.046.

[12]

J. Han and J. Jang, Nontopological bare solutions in the relativistic self-dual Maxwell-Chern-Simons-Higgs model, J. Math. Phys., 46 (2005), 042310, 16 pp. doi: 10.1063/1.1861277.

[13]

J. Han and H.-S. Nam, On the topological multivortex solutions of the self-dual Maxwell-Chern-Simons gauged O(3) sigma model, Lett. Math. Phys., 73 (2005), 17-31.  doi: 10.1007/s11005-005-8443-0.

[14]

J. Han and K. Song, Existence and asymptotic of topological solutions in the self-dual Maxwell-Chern-Simons $O(3)$ Sigma model, J. Diff. Eqn., 250 (2011), 204-222.  doi: 10.1016/j.jde.2010.08.003.

[15]

K. Kim, 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.

[16]

T. Ricciardi, Asymptotics for Maxwell-Chern-Simons multivortices, Nonlinear. Anal. TMA, 50 (2002), 1093-1106.  doi: 10.1016/S0362-546X(01)00752-0.

[17]

T. Ricciardi and G. Tarantello, Vortices in the Maxwell-Chern-Simons theory, Comm. Pure Appl. Math., 53 (2000), 811-851.  doi: 10.1002/(SICI)1097-0312(200007)53:7<811::AID-CPA2>3.0.CO;2-F.

[18]

B.-J. Schroers, Bogomol'nyi solitons in a gauged $O(3)$ sigma model, Phys. Lett. B, 356 (1995), 291-296.  doi: 10.1016/0370-2693(95)00833-7.

[19]

K. Song, Radial symmetry of topological solitons in the self-dual Maxwell-Chern-Simons $O(3)$ Sigma model, Bull. Korean. Math. Soc., 48 (2011), 1111-1117.  doi: 10.4134/BKMS.2011.48.5.1111.

[20]

Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer Monographs in Mathematics, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-6548-9.

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