-
Previous Article
A mathematical model for biodiversity diluting transmission of zika virus through competition mechanics
- DCDS-B Home
- This Issue
-
Next Article
High order one-step methods for backward stochastic differential equations via Itô-Taylor expansion
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 |
$ \times $ |
$ \begin{equation} \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)\end{equation} $ |
$ \Omega $ |
$ \mathbb{R}^2 $ |
$ a $ |
$ b $ |
$ \lambda>0 $ |
$ m_j>0(j = 1,\cdots,k_1) $ |
$ n_j>0(j = 1,\cdots,k_2) $ |
$ \delta_{p} $ |
$ p $ |
$ \Omega $ |
$ (a,b) = (0,1) $ |
$ a>b>0 $ |
$ (a,b) $ |
$ \lambda $ |
$ b>a>0 $ |
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. |
[1] |
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 |
[2] |
Youngae Lee. Topological solutions in the Maxwell-Chern-Simons model with anomalous magnetic moment. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1293-1314. doi: 10.3934/dcds.2018053 |
[3] |
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 |
[4] |
Yanqin Fang, De Tang. Method of sub-super solutions for fractional elliptic equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3153-3165. doi: 10.3934/dcdsb.2017212 |
[5] |
Kwangseok Choe, Hyungjin Huh. Chern-Simons gauged sigma model into $ \mathbb{H}^2 $ and its self-dual equations. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4613-4646. doi: 10.3934/dcds.2019189 |
[6] |
Jeongho Kim, Bora Moon. Finite difference methods for the one-dimensional Chern-Simons gauged models. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022003 |
[7] |
Jeongho Kim, Bora Moon. Hydrodynamic limits of the nonlinear Schrödinger equation with the Chern-Simons gauge fields. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2541-2561. doi: 10.3934/dcds.2021202 |
[8] |
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 |
[9] |
Youyan Wan, Jinggang Tan. The existence of nontrivial solutions to Chern-Simons-Schrödinger systems. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2765-2786. doi: 10.3934/dcds.2017119 |
[10] |
Lingyu Li, Jianfu Yang, Jinge Yang. Solutions to Chern-Simons-Schrödinger systems with external potential. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1967-1981. doi: 10.3934/dcdss.2021008 |
[11] |
Hartmut Pecher. Local solutions with infinite energy of the Maxwell-Chern-Simons-Higgs system in Lorenz gauge. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2193-2204. doi: 10.3934/dcds.2016.36.2193 |
[12] |
Jincai Kang, Chunlei Tang. Existence of nontrivial solutions to Chern-Simons-Schrödinger system with indefinite potential. Discrete and Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021016 |
[13] |
Berat Karaagac. New exact solutions for some fractional order differential equations via improved sub-equation method. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 447-454. doi: 10.3934/dcdss.2019029 |
[14] |
Hartmut Pecher. The Chern-Simons-Higgs and the Chern-Simons-Dirac equations in Fourier-Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4875-4893. doi: 10.3934/dcds.2019199 |
[15] |
Hongxia Yin. An iterative method for general variational inequalities. Journal of Industrial and Management Optimization, 2005, 1 (2) : 201-209. doi: 10.3934/jimo.2005.1.201 |
[16] |
Hyungjin Huh. Towards the Chern-Simons-Higgs equation with finite energy. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1145-1159. doi: 10.3934/dcds.2011.30.1145 |
[17] |
Nikolaos Bournaveas, Timothy Candy, Shuji Machihara. A note on the Chern-Simons-Dirac equations in the Coulomb gauge. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2693-2701. doi: 10.3934/dcds.2014.34.2693 |
[18] |
Anis Theljani, Ke Chen. An augmented lagrangian method for solving a new variational model based on gradients similarity measures and high order regulariation for multimodality registration. Inverse Problems and Imaging, 2019, 13 (2) : 309-335. doi: 10.3934/ipi.2019016 |
[19] |
Yoshifumi Aimoto, Takayasu Matsuo, Yuto Miyatake. A local discontinuous Galerkin method based on variational structure. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 817-832. doi: 10.3934/dcdss.2015.8.817 |
[20] |
Songhai Deng, Zhong Wan, Yanjiu Zhou. Optimization model and solution method for dynamically correlated two-product newsvendor problems based on Copula. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1637-1652. doi: 10.3934/dcdss.2020096 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]