June  2020, 40(6): 3291-3304. doi: 10.3934/dcds.2020127

On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory

a. 

Department of Mathematics, National Central University, Chung-Li 32001, Taiwan

b. 

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

* Corresponding author; Work partially supported by Ministry of Science and Technology of Taiwan under grant no. MOST 107-2115-M-008-005-MY3

1 Work partially supported by Ministry of Science and Technology of Taiwan under grant no. MOST 105-2115-M-008-012-MY3.
2 Work partially supported by Ministry of Science and Technology of Taiwan under grant no. MOST 106-2628-M-018-001-MY4.
Dedicated to Professor Wei-Ming Ni in honor of his 70th birthday

Received  April 2019 Published  February 2020

In this paper, by constructing a family of approximation solutions and applying a specific version of the Implicit Function Theorem (please see, e.g. [18]), we prove the existence of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory.

Citation: Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127
References:
[1]

W. Ao, C.-S. Lin and J. Wei, On non-topological solutions of the A2 and B2 Chern-Simons system, Mem. Amer. Math. Soc., 239 (2016), 1132. Google Scholar

[2]

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.  Google Scholar

[3]

D. Chae, On the elliptic system arising from a self-gravitating Born-Infeld Abelian Higgs theory, Nonlinearity, 18 (2005), 1823–1833. Google Scholar

[4]

W. Chen and C. Li, Qualitative properties of solutions to some nonlinear elliptic equations in $ {\mathbb R}^2$, Duke Math. J., 71 (1993), 427–439. doi: 10.1215/S0012-7094-93-07117-7.  Google Scholar

[5]

K.-S. Cheng and C.-S. Lin, On the conformal Gaussian curvature equation in $ {\mathbb R}^2$, J. Diff. Eqns, 146 (1998), 226–250. doi: 10.1006/jdeq.1998.3424.  Google Scholar

[6]

J.-L. Chern and S.-G. Yang, Evaluating solutions on an elliptic problem in a gravitational gauge field theory, Journal of Functional Analysis, 265 (2013), 1240-1263.   Google Scholar

[7]

J.-L. Chern and S.-G. Yang, A survey of solutions in a gravitational Born-Infeld theory, Journal of Mathematical Physics, 55 (2014), 031501, 24pp. doi: 10.1063/1.4867618.  Google Scholar

[8]

K. Choe, Uniqueness of the topological multivortex solution in the self-dual in the Chern-Simons Theorem, J. Math. Phys, 46 (2005), 012305. Google Scholar

[9]

K. Choe, N. Kim and C.-S. Lin, Existence of self-dual non-topological solutions in the Chern-Simons Higgs model, Ann.Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 28 (2011), 837–852. doi: 10.1016/j.anihpc.2011.06.003.  Google Scholar

[10]

K. ChoeN. Kim and C.-S. Lin, Self-dual symmetric nontopological solutions in the $SU(3)$ model in $ {\mathbb R}^2$, Commun. Math. Phys., 33 (2015), 1-37.   Google Scholar

[11]

K. Choe, N. Kim and C.-S. Lin, Existence of mixed type solutions in the $SU(3)$ Chern-Simons theory in $ {\mathbb R}^2$, Calc. Var. Partial Differential Equations, 56 (2017), Art. 17, 30 pp. doi: 10.1007/s00526-017-1119-7.  Google Scholar

[12]

K. ChoeN. KimY. Lin and C.-S. Lin, Existence of mixed type solutions in the Chern-Simons gauge theory of rank two in $ {\mathbb R}^2$, Journal of Functional Analysis, 273 (2017), 1734-1761.  doi: 10.1016/j.jfa.2017.05.012.  Google Scholar

[13]

A. Comtet and G. W. Gibbons, Bogomol'nyi bounds for cosmic strings, Nucl. Phys. B, 299 (1988), 719–733. doi: 10.1016/0550-3213(88)90370-7.  Google Scholar

[14]

A. V. Fursikov and O. Yu. Imanuvilov, Local exact boundary controllability of the boussinesq equation, SIAM Journal of Control and Optimization, 36 (1998), 391–421. Google Scholar

[15]

Z. Hlousek and and D. Spector, Bogomol'nyi explained, Nucl. Phys. B, 397 (1993), 173. Google Scholar

[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.  Google Scholar

[17]

B. Linet, A vortex-line model for a system of cosmic strings in equilibrium, Gen. Relativ. Gravit., 20 (1988), 451–456. doi: 10.1007/BF00758120.  Google Scholar

[18]

L. Nirenberg, Topics in Nonlinear Analysis, Courant Lecture Notes in Math., 6, American Mathematical Society, 2001. Google Scholar

[19]

D. Tong and K. Wong, Vortices and Impurities, J. High Energy Phys., 1401 (2014), 090. Google Scholar

[20]

Y. Yang, Cosmic strings in a product Abelian gauge field theory, Nucl. Phys. B, 885 (2014), 25–33. doi: 10.1016/j.nuclphysb.2014.05.013.  Google Scholar

[21]

Y. Yang, Prescribing zeros and poles on a compact Riemann surface for a gravitationally coupled Abelian gauge field theory, Comm. Math. Phys., 249 (2004), 579–609. Google Scholar

show all references

References:
[1]

W. Ao, C.-S. Lin and J. Wei, On non-topological solutions of the A2 and B2 Chern-Simons system, Mem. Amer. Math. Soc., 239 (2016), 1132. Google Scholar

[2]

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.  Google Scholar

[3]

D. Chae, On the elliptic system arising from a self-gravitating Born-Infeld Abelian Higgs theory, Nonlinearity, 18 (2005), 1823–1833. Google Scholar

[4]

W. Chen and C. Li, Qualitative properties of solutions to some nonlinear elliptic equations in $ {\mathbb R}^2$, Duke Math. J., 71 (1993), 427–439. doi: 10.1215/S0012-7094-93-07117-7.  Google Scholar

[5]

K.-S. Cheng and C.-S. Lin, On the conformal Gaussian curvature equation in $ {\mathbb R}^2$, J. Diff. Eqns, 146 (1998), 226–250. doi: 10.1006/jdeq.1998.3424.  Google Scholar

[6]

J.-L. Chern and S.-G. Yang, Evaluating solutions on an elliptic problem in a gravitational gauge field theory, Journal of Functional Analysis, 265 (2013), 1240-1263.   Google Scholar

[7]

J.-L. Chern and S.-G. Yang, A survey of solutions in a gravitational Born-Infeld theory, Journal of Mathematical Physics, 55 (2014), 031501, 24pp. doi: 10.1063/1.4867618.  Google Scholar

[8]

K. Choe, Uniqueness of the topological multivortex solution in the self-dual in the Chern-Simons Theorem, J. Math. Phys, 46 (2005), 012305. Google Scholar

[9]

K. Choe, N. Kim and C.-S. Lin, Existence of self-dual non-topological solutions in the Chern-Simons Higgs model, Ann.Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 28 (2011), 837–852. doi: 10.1016/j.anihpc.2011.06.003.  Google Scholar

[10]

K. ChoeN. Kim and C.-S. Lin, Self-dual symmetric nontopological solutions in the $SU(3)$ model in $ {\mathbb R}^2$, Commun. Math. Phys., 33 (2015), 1-37.   Google Scholar

[11]

K. Choe, N. Kim and C.-S. Lin, Existence of mixed type solutions in the $SU(3)$ Chern-Simons theory in $ {\mathbb R}^2$, Calc. Var. Partial Differential Equations, 56 (2017), Art. 17, 30 pp. doi: 10.1007/s00526-017-1119-7.  Google Scholar

[12]

K. ChoeN. KimY. Lin and C.-S. Lin, Existence of mixed type solutions in the Chern-Simons gauge theory of rank two in $ {\mathbb R}^2$, Journal of Functional Analysis, 273 (2017), 1734-1761.  doi: 10.1016/j.jfa.2017.05.012.  Google Scholar

[13]

A. Comtet and G. W. Gibbons, Bogomol'nyi bounds for cosmic strings, Nucl. Phys. B, 299 (1988), 719–733. doi: 10.1016/0550-3213(88)90370-7.  Google Scholar

[14]

A. V. Fursikov and O. Yu. Imanuvilov, Local exact boundary controllability of the boussinesq equation, SIAM Journal of Control and Optimization, 36 (1998), 391–421. Google Scholar

[15]

Z. Hlousek and and D. Spector, Bogomol'nyi explained, Nucl. Phys. B, 397 (1993), 173. Google Scholar

[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.  Google Scholar

[17]

B. Linet, A vortex-line model for a system of cosmic strings in equilibrium, Gen. Relativ. Gravit., 20 (1988), 451–456. doi: 10.1007/BF00758120.  Google Scholar

[18]

L. Nirenberg, Topics in Nonlinear Analysis, Courant Lecture Notes in Math., 6, American Mathematical Society, 2001. Google Scholar

[19]

D. Tong and K. Wong, Vortices and Impurities, J. High Energy Phys., 1401 (2014), 090. Google Scholar

[20]

Y. Yang, Cosmic strings in a product Abelian gauge field theory, Nucl. Phys. B, 885 (2014), 25–33. doi: 10.1016/j.nuclphysb.2014.05.013.  Google Scholar

[21]

Y. Yang, Prescribing zeros and poles on a compact Riemann surface for a gravitationally coupled Abelian gauge field theory, Comm. Math. Phys., 249 (2004), 579–609. Google Scholar

[1]

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

[2]

Feng Li, Yuxiang Li. Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5409-5436. doi: 10.3934/dcdsb.2019064

[3]

Anupam Sen, T. Raja Sekhar. Structural stability of the Riemann solution for a strictly hyperbolic system of conservation laws with flux approximation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 931-942. doi: 10.3934/cpaa.2019045

[4]

Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213

[5]

Jean-Jérôme Casanova. Existence of time-periodic strong solutions to a fluid–structure system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3291-3313. doi: 10.3934/dcds.2019136

[6]

Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041

[7]

Francisco Ortegón Gallego, María Teresa González Montesinos. Existence of a capacity solution to a coupled nonlinear parabolic--elliptic system. Communications on Pure & Applied Analysis, 2007, 6 (1) : 23-42. doi: 10.3934/cpaa.2007.6.23

[8]

Thi-Bich-Ngoc Mac. Existence of solution for a system of repulsion and alignment: Comparison between theory and simulation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3013-3027. doi: 10.3934/dcdsb.2015.20.3013

[9]

Yong Zeng. Existence and uniqueness of very weak solution of the MHD type system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 5617-5638. doi: 10.3934/dcds.2020240

[10]

Jitsuro Sugie, Tadayuki Hara. Existence and non-existence of homoclinic trajectories of the Liénard system. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 237-254. doi: 10.3934/dcds.1996.2.237

[11]

Vicent Caselles. An existence and uniqueness result for flux limited diffusion equations. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1151-1195. doi: 10.3934/dcds.2011.31.1151

[12]

Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125

[13]

Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure & Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011

[14]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[15]

Chunxiao Guo, Fan Cui, Yongqian Han. Global existence and uniqueness of the solution for the fractional Schrödinger-KdV-Burgers system. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1687-1699. doi: 10.3934/dcdss.2016070

[16]

Masaki Kurokiba, Toshitaka Nagai, T. Ogawa. The uniform boundedness and threshold for the global existence of the radial solution to a drift-diffusion system. Communications on Pure & Applied Analysis, 2006, 5 (1) : 97-106. doi: 10.3934/cpaa.2006.5.97

[17]

Zhaoquan Xu, Jiying Ma. Monotonicity, asymptotics and uniqueness of travelling wave solution of a non-local delayed lattice dynamical system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5107-5131. doi: 10.3934/dcds.2015.35.5107

[18]

Darko Mitrovic. Existence and stability of a multidimensional scalar conservation law with discontinuous flux. Networks & Heterogeneous Media, 2010, 5 (1) : 163-188. doi: 10.3934/nhm.2010.5.163

[19]

Guangyu Xu, Chunlai Mu, Dan Li. Global existence and non-existence analyses to a nonlinear Klein-Gordon system with damping terms under positive initial energy. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2491-2512. doi: 10.3934/cpaa.2020109

[20]

Sebastian Aniţa, William Edward Fitzgibbon, Michel Langlais. Global existence and internal stabilization for a reaction-diffusion system posed on non coincident spatial domains. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 805-822. doi: 10.3934/dcdsb.2009.11.805

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (66)
  • HTML views (137)
  • Cited by (0)

[Back to Top]