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August  2022, 27(8): 4415-4428. doi: 10.3934/dcdsb.2021234

## Vortex Condensation in General U(1)×U(1) Abelian Chern-Simons Model on a flat torus

 Department of Applied Mathematics, National Yang Ming Chiao Tung University, Taiwan, National Center for Theoretical Sciences, National Taiwan University, Taipei, Taiwan

* Corresponding author: Hsin-Yuan Huang

Received  January 2021 Revised  July 2021 Published  August 2022 Early access  September 2021

Fund Project: The author is partially supported by Ministry of Science and Technology of Taiwan under the grant MOST 106-2628-M-009-005-MY4

In this paper, we study an elliptic system arising from the U(1)
 $\times$
U(1) Abelian Chern-Simons Model[25,37] of the form
 $$$\left\{\begin{split} \Delta u = &\lambda \left(a(b-a)e^{u}-b(b-a)e^{v}+a^2e^{2u} -abe^{2v}+b(b-a)e^{u+v}\right)\\ & +4\pi \sum\limits_{j = 1}^{k_1}m_j\delta_{p_j}, \\ \Delta v = &\lambda \left(-b(b-a)e^{u}+a(b-a)e^{v}-abe^{2u} +a^2e^{2v}+b(b-a)e^{u+v}\right)\\ & +4\pi \sum\limits_{j = 1}^{k_2}n_j\delta_{q_j}, \end{split}\right. \quad\quad\quad\quad (1)$$$
which are defined on a parallelogram
 $\Omega$
in
 $\mathbb{R}^2$
with doubly periodic boundary conditions. Here,
 $a$
and
 $b$
are interaction constants,
 $\lambda>0$
is related to coupling constant,
 $m_j>0(j = 1,\cdots,k_1)$
,
 $n_j>0(j = 1,\cdots,k_2)$
,
 $\delta_{p}$
is the Dirac measure,
 $p$
is called vortex point. Concerning the existence results of this system over
 $\Omega$
, only the cases
 $(a,b) = (0,1)$
[28] and
 $a>b>0$
[14] were studied in the literature. The solvability of this system (1) is still an open problem as regards other parameters
 $(a,b)$
. We show that the system (1) admits topological solutions provided
 $\lambda$
is large and
 $b>a>0$
Our arguments are based on a iteration scheme and variational formulation.
Citation: Hsin-Yuan Huang. Vortex Condensation in General U(1)×U(1) Abelian Chern-Simons Model on a flat torus. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4415-4428. doi: 10.3934/dcdsb.2021234
##### References:
 [1] A. A. Abrikosov, The magnetic properties of superconducting alloys, J. Phys. Chem. Solids, 2 (1957), 199-208.  doi: 10.1016/0022-3697(57)90083-5. [2] T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations, volume 252 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5734-9. [3] E. B. Bogomol'n$\check{\mathrm{y}}$, The stability of classical solutions, Jadernaja Fiz., 24 (1976), 861-870. [4] L. A. Caffarelli and Y. Yang, Vortex condensation in the Chern-Simons Higgs model: An existence theorem, Comm. Math. Phys., 168 (1995), 321-336.  doi: 10.1007/BF02101552. [5] C.-C. Chen, C.-S. Lin and G. Wang, Concentration phenomena of two-vortex solutions in a Chern-Simons model, Ann. Sc. Norm. Super. Pisa., 3 (2004), 367-397. [6] J.-L. Chern, Z.-Y. Chen and C.-S. Lin, Uniqueness of topological solutions and the structure of solutions for the Chern-Simons system with two Higgs particles, Comm. Math. Phys., 296 (2010), 323-351.  doi: 10.1007/s00220-010-1021-z. [7] 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. [8] K. Choe, N. Kim and C.-S. Lin, Existence of radial mixed type solutions in Chern–Simons theories of rank 2 in $\mathbb{R}^2$, Comm. Math. Phys., 370 (2019), 995-1017.  doi: 10.1007/s00220-019-03469-6. [9] M. del Pino, P. Esposito, P. Figueroa and M. Musso, Nontopological condensates for the self-dual Chern-Simons-Higgs model, Comm. Pure Appl. Math., 68 (2015), 1191-1283.  doi: 10.1002/cpa.21548. [10] J. Dziarmaga, Low energy dynamics of ${[\mathrm{U}(1)]}^{N}$ chern-simons solitons, Phys. Rev. D, 49 (1994), 5469-5479. [11] J. Fröhlich and P. A. Marchetti, Quantum field theory of anyons, Lett. Math. Phys., 16 (1988), 347-358.  doi: 10.1007/BF00402043. [12] J. Fröhlich and P. A. Marchetti, Quantum field theories of vortices and anyons, Comm. Math. Phys., 121 (1989), 177-223.  doi: 10.1007/BF01217803. [13] X. Han, H.-Y. Huang and C.-S. Lin, Bubbling solutions for a skew-symmetric Chern-Simons system in a torus, J. Funct. Anal., 273 (2017), 1354-1396.  doi: 10.1016/j.jfa.2017.04.018. [14] X. Han and G. Tarantello, Doubly periodic self-dual vortices in a relativistic non-abelian Chern–Simons model, Calc. Var. Partial Differ. Equ., 49 (2014), 1149-1176.  doi: 10.1007/s00526-013-0615-7. [15] J. Hong, Y. Kim and P. Y. Pac, Multivortex solutions of the abelian Chern-Simons-Higgs theory, Phys. Rev. Lett., 64 (1990), 2230-2233.  doi: 10.1103/PhysRevLett.64.2230. [16] G. Huang and C.-S. Lin, The existence of non-topological solutions for a skew-symmetric Chern-Simons system, Indiana Univ. Math. J., 65 (2016), 453-491.  doi: 10.1512/iumj.2016.65.5769. [17] H.-Y. Huang, Y. Lee and C.-S. Lin, Uniqueness of topological multi-vortex solutions for a skew-symmetric Chern-Simons system, J. Math. Phys., 56 (2015), 041501, 12 pp. doi: 10.1063/1.4916290. [18] H.-Y. Huang and C.-S. Lin, Uniqueness of non-topological solutions for the Chern-Simons system with two Higgs particles, Kodai Math. J., 37 (2014), 274-284.  doi: 10.2996/kmj/1404393887. [19] H.-Y. Huang and C.-S. Lin, Classification of the entire radial self-dual solutions to non-abelian Chern–Simons systems, J. Funct. Anal., 266 (2014), 6796-6841.  doi: 10.1016/j.jfa.2014.03.007. [20] H.-Y. Huang and L. Zhang, The domain geometry and the bubbling phenomenon of rank two gauge theory, Comm. Math. Phys., 349 (2017), 393-424.  doi: 10.1007/s00220-016-2685-9. [21] R. Jackiw and E. J. Weinberg, Self-dual Chern-Simons vortices, Phys. Rev. Lett., 64 (1990), 2234-2237.  doi: 10.1103/PhysRevLett.64.2234. [22] A. Jaffe and C. Taubes, Vortices and Monopoles: Structure of Static Gauge Theories, Birkhäuser, Boston, Mass., 1980. [23] B. Julia and A. Zee, Poles with both magnetic and electric charges in non-abelian gauge theory, Physical Review D, 11 (1975), 2227.  doi: 10.1103/PhysRevD.11.2227. [24] D. I. Khomskii and A. Freimuth, Charged vortices in high temperature superconductors, Physical Review Letters, 75 (1995), 1384.  doi: 10.1103/PhysRevLett.75.1384. [25] C. Kim, C. Lee, P. Ko, B.-H. Lee and H. Min, Schrödinger fields on the plane with ${[\mathrm{U}(1)]}^{N}$ Chern-Simons interactions and generalized self-dual solitons, Phys. Rev. D, 48 (1993), 1821-1840.  doi: 10.1103/PhysRevD.48.1821. [26] K. Kumagai, K. Nozaki and Y. Matsuda, Charged vortices in high-temperature superconductors probed by NMR, Phys. Rev. B, 63 (2001), 144502.  doi: 10.1103/PhysRevB.63.144502. [27] C.-S. Lin, A. C. Ponce and Y. Yang, A system of elliptic equations arising in Chern-Simons field theory, J. Funct. Anal., 247 (2007), 289-350.  doi: 10.1016/j.jfa.2007.03.010. [28] C.-S. Lin and J. V. Prajapat, Vortex condensates for relativistic abelian Chern-Simons model with two Higgs scalar fields and two gauge fields on a torus, Comm. Math. Phys., 288 (2009), 311-347.  doi: 10.1007/s00220-009-0774-8. [29] C.-S. Lin and S. Yan, Existence of bubbling solutions for Chern-Simons model on a torus, Arc. Ration. Mech. Anal., 207 (2013), 353-392.  doi: 10.1007/s00205-012-0575-7. [30] C.-S. Lin and S. Yan, On condensate of solutions for the Chern-Simons-Higgs equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 1329-1354.  doi: 10.1016/j.anihpc.2016.10.006. [31] C.-S. Lin and S. Yan, On the mean field type bubbling solutions for Chern-Simons-Higgs equation, Adv. Math., 338 (2018), 1141-1188.  doi: 10.1016/j.aim.2018.09.021. [32] A. Poliakovsky and G. Tarantello, On non-topological solutions for planar liouville systems of Toda-type, Comm. Math. Phys., 347 (2016), 223-270.  doi: 10.1007/s00220-016-2662-3. [33] M. Prasad and C. Sommerfield, Exact classical solution for the't hooft monopole and the Julia-Zee dyon, Phys. Rev. Lett., 35 (1975), 760. [34] L. H. Ryder, Quantum Field Theory, Cambridge university press, 1996.  doi: 10.1017/CBO9780511813900. [35] J. Spruck and Y. Yang, The existence of nontopological solitons in the self-dual Chern-Simons theory, Comm. Math. Phys., 149 (1992), 361-376.  doi: 10.1007/BF02097630. [36] G. Tarantello, Uniqueness of selfdual periodic Chern-Simons vortices of topological-type, Calc. Var. Partial Differ. Equ., 29 (2007), 191-217.  doi: 10.1007/s00526-006-0062-9. [37] F. Wilczek, Disassembling anyons, Phys. Rev. Lett., 69 (1992), 132-135.  doi: 10.1103/PhysRevLett.69.132.

show all references

##### References:
 [1] A. A. Abrikosov, The magnetic properties of superconducting alloys, J. Phys. Chem. Solids, 2 (1957), 199-208.  doi: 10.1016/0022-3697(57)90083-5. [2] T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations, volume 252 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5734-9. [3] E. B. Bogomol'n$\check{\mathrm{y}}$, The stability of classical solutions, Jadernaja Fiz., 24 (1976), 861-870. [4] L. A. Caffarelli and Y. Yang, Vortex condensation in the Chern-Simons Higgs model: An existence theorem, Comm. Math. Phys., 168 (1995), 321-336.  doi: 10.1007/BF02101552. [5] C.-C. Chen, C.-S. Lin and G. Wang, Concentration phenomena of two-vortex solutions in a Chern-Simons model, Ann. Sc. Norm. Super. Pisa., 3 (2004), 367-397. [6] J.-L. Chern, Z.-Y. Chen and C.-S. Lin, Uniqueness of topological solutions and the structure of solutions for the Chern-Simons system with two Higgs particles, Comm. Math. Phys., 296 (2010), 323-351.  doi: 10.1007/s00220-010-1021-z. [7] 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. [8] K. Choe, N. Kim and C.-S. Lin, Existence of radial mixed type solutions in Chern–Simons theories of rank 2 in $\mathbb{R}^2$, Comm. Math. Phys., 370 (2019), 995-1017.  doi: 10.1007/s00220-019-03469-6. [9] M. del Pino, P. Esposito, P. Figueroa and M. Musso, Nontopological condensates for the self-dual Chern-Simons-Higgs model, Comm. Pure Appl. Math., 68 (2015), 1191-1283.  doi: 10.1002/cpa.21548. [10] J. Dziarmaga, Low energy dynamics of ${[\mathrm{U}(1)]}^{N}$ chern-simons solitons, Phys. Rev. D, 49 (1994), 5469-5479. [11] J. Fröhlich and P. A. Marchetti, Quantum field theory of anyons, Lett. Math. Phys., 16 (1988), 347-358.  doi: 10.1007/BF00402043. [12] J. Fröhlich and P. A. Marchetti, Quantum field theories of vortices and anyons, Comm. Math. Phys., 121 (1989), 177-223.  doi: 10.1007/BF01217803. [13] X. Han, H.-Y. Huang and C.-S. Lin, Bubbling solutions for a skew-symmetric Chern-Simons system in a torus, J. Funct. Anal., 273 (2017), 1354-1396.  doi: 10.1016/j.jfa.2017.04.018. [14] X. Han and G. Tarantello, Doubly periodic self-dual vortices in a relativistic non-abelian Chern–Simons model, Calc. Var. Partial Differ. Equ., 49 (2014), 1149-1176.  doi: 10.1007/s00526-013-0615-7. [15] J. Hong, Y. Kim and P. Y. Pac, Multivortex solutions of the abelian Chern-Simons-Higgs theory, Phys. Rev. Lett., 64 (1990), 2230-2233.  doi: 10.1103/PhysRevLett.64.2230. [16] G. Huang and C.-S. Lin, The existence of non-topological solutions for a skew-symmetric Chern-Simons system, Indiana Univ. Math. J., 65 (2016), 453-491.  doi: 10.1512/iumj.2016.65.5769. [17] H.-Y. Huang, Y. Lee and C.-S. Lin, Uniqueness of topological multi-vortex solutions for a skew-symmetric Chern-Simons system, J. Math. Phys., 56 (2015), 041501, 12 pp. doi: 10.1063/1.4916290. [18] H.-Y. Huang and C.-S. Lin, Uniqueness of non-topological solutions for the Chern-Simons system with two Higgs particles, Kodai Math. J., 37 (2014), 274-284.  doi: 10.2996/kmj/1404393887. [19] H.-Y. Huang and C.-S. Lin, Classification of the entire radial self-dual solutions to non-abelian Chern–Simons systems, J. Funct. Anal., 266 (2014), 6796-6841.  doi: 10.1016/j.jfa.2014.03.007. [20] H.-Y. Huang and L. Zhang, The domain geometry and the bubbling phenomenon of rank two gauge theory, Comm. Math. Phys., 349 (2017), 393-424.  doi: 10.1007/s00220-016-2685-9. [21] R. Jackiw and E. J. Weinberg, Self-dual Chern-Simons vortices, Phys. Rev. Lett., 64 (1990), 2234-2237.  doi: 10.1103/PhysRevLett.64.2234. [22] A. Jaffe and C. Taubes, Vortices and Monopoles: Structure of Static Gauge Theories, Birkhäuser, Boston, Mass., 1980. [23] B. Julia and A. Zee, Poles with both magnetic and electric charges in non-abelian gauge theory, Physical Review D, 11 (1975), 2227.  doi: 10.1103/PhysRevD.11.2227. [24] D. I. Khomskii and A. Freimuth, Charged vortices in high temperature superconductors, Physical Review Letters, 75 (1995), 1384.  doi: 10.1103/PhysRevLett.75.1384. [25] C. Kim, C. Lee, P. Ko, B.-H. Lee and H. Min, Schrödinger fields on the plane with ${[\mathrm{U}(1)]}^{N}$ Chern-Simons interactions and generalized self-dual solitons, Phys. Rev. D, 48 (1993), 1821-1840.  doi: 10.1103/PhysRevD.48.1821. [26] K. Kumagai, K. Nozaki and Y. Matsuda, Charged vortices in high-temperature superconductors probed by NMR, Phys. Rev. B, 63 (2001), 144502.  doi: 10.1103/PhysRevB.63.144502. [27] C.-S. Lin, A. C. Ponce and Y. Yang, A system of elliptic equations arising in Chern-Simons field theory, J. Funct. Anal., 247 (2007), 289-350.  doi: 10.1016/j.jfa.2007.03.010. [28] C.-S. Lin and J. V. Prajapat, Vortex condensates for relativistic abelian Chern-Simons model with two Higgs scalar fields and two gauge fields on a torus, Comm. Math. Phys., 288 (2009), 311-347.  doi: 10.1007/s00220-009-0774-8. [29] C.-S. Lin and S. Yan, Existence of bubbling solutions for Chern-Simons model on a torus, Arc. Ration. Mech. Anal., 207 (2013), 353-392.  doi: 10.1007/s00205-012-0575-7. [30] C.-S. Lin and S. Yan, On condensate of solutions for the Chern-Simons-Higgs equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 1329-1354.  doi: 10.1016/j.anihpc.2016.10.006. [31] C.-S. Lin and S. Yan, On the mean field type bubbling solutions for Chern-Simons-Higgs equation, Adv. Math., 338 (2018), 1141-1188.  doi: 10.1016/j.aim.2018.09.021. [32] A. Poliakovsky and G. Tarantello, On non-topological solutions for planar liouville systems of Toda-type, Comm. Math. Phys., 347 (2016), 223-270.  doi: 10.1007/s00220-016-2662-3. [33] M. Prasad and C. Sommerfield, Exact classical solution for the't hooft monopole and the Julia-Zee dyon, Phys. Rev. Lett., 35 (1975), 760. [34] L. H. Ryder, Quantum Field Theory, Cambridge university press, 1996.  doi: 10.1017/CBO9780511813900. [35] J. Spruck and Y. Yang, The existence of nontopological solitons in the self-dual Chern-Simons theory, Comm. Math. Phys., 149 (1992), 361-376.  doi: 10.1007/BF02097630. [36] G. Tarantello, Uniqueness of selfdual periodic Chern-Simons vortices of topological-type, Calc. Var. Partial Differ. Equ., 29 (2007), 191-217.  doi: 10.1007/s00526-006-0062-9. [37] F. Wilczek, Disassembling anyons, Phys. Rev. Lett., 69 (1992), 132-135.  doi: 10.1103/PhysRevLett.69.132.
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