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July  2017, 16(4): 1315-1330. doi: 10.3934/cpaa.2017064

Non-topological solutions in a generalized Chern-Simons model on torus

National Institute for Mathematical Sciences, Academic exchanges, KT Daeduk 2 Research Center, 70 Yuseong-daero 1689 beon-gil, Yuseong-gu, Daejeon, 34047, Republic of Korea

* Corresponding author

Received  August 2016 Revised  February 2017 Published  April 2017

We consider a quasi-linear elliptic equation with Dirac source terms arising in a generalized self-dual Chern-Simons-Higgs gauge theory. In this paper, we study doubly periodic vortices with arbitrary vortex configuration. First of all, we show that under doubly periodic condition, there are only two types of solutions, topological and non-topological solutions as the coupling parameter goes to zero. Moreover, we succeed to construct non-topological solution with $k$ bubbles where $k\in\mathbb{N}$ is any given number. To find a solution, we analyze the structure of quasi-linear elliptic equation carefully and apply the method developed in the recent work [16].

Citation: Youngae Lee. Non-topological solutions in a generalized Chern-Simons model on torus. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1315-1330. doi: 10.3934/cpaa.2017064
References:
[1]

J. BurzlaffA. Chakrabarti and D. H. Tchrakian, Generalized self-dual Chern-Simons vortices, Phys. Lett. B, 293 (1992), 127-131.  doi: 10.1016/0370-2693(92)91490-Z.

[2]

L. A. Caffarelli and Y. S. Yang, Vortex condensation in the Chern-Simons Higgs model: an existence theorem, Comm. Math. Phys., 168 (1995), 321-336. 

[3]

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), 119-142.  doi: 10.1007/s002200000302.

[4]

D. Chae and O. Y. Imanuvilov, Non-topological solutions in the generalized self-dual ChernSimons-Higgs theory, Calc. Var. Partial Differential Equations, 16 (2003), 47-61.  doi: 10.1007/s005260100141.

[5]

H. ChanC. C. Fu and C. S. Lin, Non-topological multivortex solutions to the self-dual Chern-Simons-Higgs equation, Comm. Math. Phys., 231 (2002), 189-221.  doi: 10.1007/s00220-002-0691-6.

[6]

K. Choe, Uniqueness of the topological multivortex solution in the self-dual Chern-Simons theory, J. Math. Phys., 46 (2005), 012305, 22 pp.  doi: 10.1063/1.1834694.

[7]

K. Choe, Asymptotic behavior of condensate solutions in the Chern-Simons-Higgs theory, J. Math. Phys., 48 (2007), 48 (2007), 103501, 17 pp.  doi: 10.1063/1.2785821.

[8]

K. Choe and N. Kim, Blow-up solutions of the self-dual Chern-Simons-Higgs vortex equation, Ann. Inst. H. Poincaré Anal. Non Linaire, 25 (2008), 313-338.  doi: 10.1016/j.anihpc.2006.11.012.

[9]

W. DingJ. JostJ. LiX. Peng and G. Wang, Self duality equations for Ginzburg-Landau and Seiberg-Witten type functionals with 6th order potentials, Comm. Math. Phys., 217 (2001), 383-407.  doi: 10.1007/s002200100377.

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 224 second ed. , Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[11]

X. Han, Existence of doubly periodic vortices in a generalized Chern-Simons model, Nonlinear Anal. Real World Appl., 16 (2014), 90-102.  doi: 10.1016/j.nonrwa.2013.09.009.

[12]

J. HongY. Kim and P. Y. Pac, Multi-vortex solutions of the Abelian Chern-Simons-Higgs theory, Phys. Rev. Lett., 64 (1990), 2230-2233.  doi: 10.1103/PhysRevLett.64.2230.

[13]

R. Jackiw and E. J. Weinberg, Self-dual Chern-Simons vortices, Phys. Rev. Lett., 64 (1990), 2234-2237.  doi: 10.1103/PhysRevLett.64.2234.

[14]

A. Jaffe and C. Taubes, Vortices and Monopoles, Birkhäuser, Boston, 1980.

[15]

C. S. Lin and S. Yan, Bubbling solutions for relativistic abelian Chern-Simons model on a torus, Comm. Math. Phys., 297 (2010), 733-758.  doi: 10.1007/s00220-010-1056-1.

[16]

C. S. Lin and S. Yan, Existence of Bubbling solutions for Chern-Simons model on a torus, Arch. Ration. Mech. Anal., 207 (2013), 353-392.  doi: 10.1007/s00205-012-0575-7.

[17]

M. Nolasco and G. Tarantello, On a sharp Sobolev-type inequality on two dimensional compact manifolds, Arch. Ration. Mech. Anal., 145 (1998), 161-195.  doi: 10.1007/s002050050127.

[18]

M. Nolasco and G. Tarantello, Double vortex condensates in the Chern-Simons-Higgs theory, Calc. Var. Partial Differential Equations, 9 (1999), 31-94.  doi: 10.1007/s005260050132.

[19]

J. Spruck and Y. Yang, Topological solutions in the self-dual Chern-Simons theory: existence and approximation, Ann. Inst. H. Poincare Anal. Non Lineaire, 12 (1995), 75-97. 

[20]

G. 't Hooft, A property of electric and magnetic flux in nonabelian gauge theories, Nucl. Phys. B, 153 (1979), 141-160.  doi: 10.1016/0550-3213(79)90465-6.

[21]

G. Tarantello, Multiple condensate solutions for the Chern-Simons-Higgs theory, J. Math. Phys., 37 (1996), 3769-3796.  doi: 10.1063/1.531601.

[22]

G. Tarantello, Selfdual Gauge Field Vortices. An Analytical Approach, Progress in Nonlinear Differential Equations and their Applications. Birkhauser Boston, Inc. , Boston, 2008. doi: 10.1007/978-0-8176-4608-0.

[23]

D. H. Tchrakian and Y. Yang, The existence of generalised self-dual Chern-Simons vortices, Lett. Math. Phys., 36 (1996), 403-413.  doi: 10.1007/BF00714405.

[24]

Y. Yang, Chern-Simons solitons and a nonlinear elliptic equation, Helv. Phys. Acta, 71 (1998), 573-585. 

[25]

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]

J. BurzlaffA. Chakrabarti and D. H. Tchrakian, Generalized self-dual Chern-Simons vortices, Phys. Lett. B, 293 (1992), 127-131.  doi: 10.1016/0370-2693(92)91490-Z.

[2]

L. A. Caffarelli and Y. S. Yang, Vortex condensation in the Chern-Simons Higgs model: an existence theorem, Comm. Math. Phys., 168 (1995), 321-336. 

[3]

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), 119-142.  doi: 10.1007/s002200000302.

[4]

D. Chae and O. Y. Imanuvilov, Non-topological solutions in the generalized self-dual ChernSimons-Higgs theory, Calc. Var. Partial Differential Equations, 16 (2003), 47-61.  doi: 10.1007/s005260100141.

[5]

H. ChanC. C. Fu and C. S. Lin, Non-topological multivortex solutions to the self-dual Chern-Simons-Higgs equation, Comm. Math. Phys., 231 (2002), 189-221.  doi: 10.1007/s00220-002-0691-6.

[6]

K. Choe, Uniqueness of the topological multivortex solution in the self-dual Chern-Simons theory, J. Math. Phys., 46 (2005), 012305, 22 pp.  doi: 10.1063/1.1834694.

[7]

K. Choe, Asymptotic behavior of condensate solutions in the Chern-Simons-Higgs theory, J. Math. Phys., 48 (2007), 48 (2007), 103501, 17 pp.  doi: 10.1063/1.2785821.

[8]

K. Choe and N. Kim, Blow-up solutions of the self-dual Chern-Simons-Higgs vortex equation, Ann. Inst. H. Poincaré Anal. Non Linaire, 25 (2008), 313-338.  doi: 10.1016/j.anihpc.2006.11.012.

[9]

W. DingJ. JostJ. LiX. Peng and G. Wang, Self duality equations for Ginzburg-Landau and Seiberg-Witten type functionals with 6th order potentials, Comm. Math. Phys., 217 (2001), 383-407.  doi: 10.1007/s002200100377.

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 224 second ed. , Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[11]

X. Han, Existence of doubly periodic vortices in a generalized Chern-Simons model, Nonlinear Anal. Real World Appl., 16 (2014), 90-102.  doi: 10.1016/j.nonrwa.2013.09.009.

[12]

J. HongY. Kim and P. Y. Pac, Multi-vortex solutions of the Abelian Chern-Simons-Higgs theory, Phys. Rev. Lett., 64 (1990), 2230-2233.  doi: 10.1103/PhysRevLett.64.2230.

[13]

R. Jackiw and E. J. Weinberg, Self-dual Chern-Simons vortices, Phys. Rev. Lett., 64 (1990), 2234-2237.  doi: 10.1103/PhysRevLett.64.2234.

[14]

A. Jaffe and C. Taubes, Vortices and Monopoles, Birkhäuser, Boston, 1980.

[15]

C. S. Lin and S. Yan, Bubbling solutions for relativistic abelian Chern-Simons model on a torus, Comm. Math. Phys., 297 (2010), 733-758.  doi: 10.1007/s00220-010-1056-1.

[16]

C. S. Lin and S. Yan, Existence of Bubbling solutions for Chern-Simons model on a torus, Arch. Ration. Mech. Anal., 207 (2013), 353-392.  doi: 10.1007/s00205-012-0575-7.

[17]

M. Nolasco and G. Tarantello, On a sharp Sobolev-type inequality on two dimensional compact manifolds, Arch. Ration. Mech. Anal., 145 (1998), 161-195.  doi: 10.1007/s002050050127.

[18]

M. Nolasco and G. Tarantello, Double vortex condensates in the Chern-Simons-Higgs theory, Calc. Var. Partial Differential Equations, 9 (1999), 31-94.  doi: 10.1007/s005260050132.

[19]

J. Spruck and Y. Yang, Topological solutions in the self-dual Chern-Simons theory: existence and approximation, Ann. Inst. H. Poincare Anal. Non Lineaire, 12 (1995), 75-97. 

[20]

G. 't Hooft, A property of electric and magnetic flux in nonabelian gauge theories, Nucl. Phys. B, 153 (1979), 141-160.  doi: 10.1016/0550-3213(79)90465-6.

[21]

G. Tarantello, Multiple condensate solutions for the Chern-Simons-Higgs theory, J. Math. Phys., 37 (1996), 3769-3796.  doi: 10.1063/1.531601.

[22]

G. Tarantello, Selfdual Gauge Field Vortices. An Analytical Approach, Progress in Nonlinear Differential Equations and their Applications. Birkhauser Boston, Inc. , Boston, 2008. doi: 10.1007/978-0-8176-4608-0.

[23]

D. H. Tchrakian and Y. Yang, The existence of generalised self-dual Chern-Simons vortices, Lett. Math. Phys., 36 (1996), 403-413.  doi: 10.1007/BF00714405.

[24]

Y. Yang, Chern-Simons solitons and a nonlinear elliptic equation, Helv. Phys. Acta, 71 (1998), 573-585. 

[25]

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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