American Institute of Mathematical Sciences

Advanced
June  2005, 4(2): 283-294. doi: 10.3934/cpaa.2005.4.283

On positive solutions of the elliptic sine-Gordon equation

Goong Chen 1, , Zhonghai Ding 2, and  Shujie Li 3,

1. 

Department of Mathematics, Texas A&M University, College Station, TX 77843, United States
2. 

Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV 89154-4020, United States
3. 

Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 100080

Received  April 2004 Revised  November 2004 Published  March 2005

The aim of this paper is to study positive solutions of the elliptic sine-Gordon equation on a bounded domain with homogeneous Dirichlet boundary condition, which models the steady state of the Josephson $\pi-$junction in superconductivity. The properties of positive solutions are investigated theoretically and numerically.
Keywords: variational methods, positive solutions., Elliptic sine-Gordon equation.
Mathematics Subject Classification: Primary:35J20,58E05;Secondary:35Q20,65N3.
Citation: Goong Chen, Zhonghai Ding, Shujie Li. On positive solutions of the elliptic sine-Gordon equation. Communications on Pure & Applied Analysis, 2005, 4 (2) : 283-294. doi: 10.3934/cpaa.2005.4.283
[1]

Christopher K. R. T. Jones, Robert Marangell. The spectrum of travelling wave solutions to the Sine-Gordon equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 925-937. doi: 10.3934/dcdss.2012.5.925
[2]

Cornelia Schiebold. Noncommutative AKNS systems and multisoliton solutions to the matrix sine-gordon equation. Conference Publications, 2009, 2009 (Special) : 678-690. doi: 10.3934/proc.2009.2009.678
[3]

Qin Sheng, David A. Voss, Q. M. Khaliq. An adaptive splitting algorithm for the sine-Gordon equation. Conference Publications, 2005, 2005 (Special) : 792-797. doi: 10.3934/proc.2005.2005.792
[4]

Carl-Friedrich Kreiner, Johannes Zimmer. Heteroclinic travelling waves for the lattice sine-Gordon equation with linear pair interaction. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 915-931. doi: 10.3934/dcds.2009.25.915
[5]

Shuang Yang, Yangrong Li. Forward controllability of a random attractor for the non-autonomous stochastic sine-Gordon equation on an unbounded domain. Evolution Equations & Control Theory, 2020, 9 (3) : 581-604. doi: 10.3934/eect.2020025
[6]

Sara Cuenda, Niurka R. Quintero, Angel Sánchez. Sine-Gordon wobbles through Bäcklund transformations. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1047-1056. doi: 10.3934/dcdss.2011.4.1047
[7]

Igor Chueshov, Peter E. Kloeden, Meihua Yang. Synchronization in coupled stochastic sine-Gordon wave model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2969-2990. doi: 10.3934/dcdsb.2016082
[8]

Ivan Christov, C. I. Christov. The coarse-grain description of interacting sine-Gordon solitons with varying widths. Conference Publications, 2009, 2009 (Special) : 171-180. doi: 10.3934/proc.2009.2009.171
[9]

Yangrong Li, Shuang Yang. Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1155-1175. doi: 10.3934/cpaa.2019056
[10]

V. V. Chepyzhov, M. I. Vishik, W. L. Wendland. On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 27-38. doi: 10.3934/dcds.2005.12.27
[11]

Ruofei Yao, Yi Li, Hongbin Chen. Uniqueness of positive radial solutions of a semilinear elliptic equation in an annulus. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1585-1594. doi: 10.3934/dcds.2018122
[12]

Fang-Fang Liao, Chun-Lei Tang. Four positive solutions of a quasilinear elliptic equation in $ R^N$. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2577-2600. doi: 10.3934/cpaa.2013.12.2577
[13]

Giovany M. Figueiredo, Tarcyana S. Figueiredo-Sousa, Cristian Morales-Rodrigo, Antonio Suárez. Existence of positive solutions of an elliptic equation with local and nonlocal variable diffusion coefficient. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3689-3711. doi: 10.3934/dcdsb.2018311
[14]

Xudong Shang, Jihui Zhang. Multi-peak positive solutions for a fractional nonlinear elliptic equation. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 3183-3201. doi: 10.3934/dcds.2015.35.3183
[15]

Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179
[16]

Yinbin Deng, Qi Gao. Asymptotic behavior of the positive solutions for an elliptic equation with Hardy term. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 367-380. doi: 10.3934/dcds.2009.24.367
[17]

Zaihui Gan. Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1541-1554. doi: 10.3934/cpaa.2009.8.1541
[18]

Raffaella Servadei, Enrico Valdinoci. Variational methods for non-local operators of elliptic type. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 2105-2137. doi: 10.3934/dcds.2013.33.2105
[19]

Andrea Malchiodi. Topological methods for an elliptic equation with exponential nonlinearities. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 277-294. doi: 10.3934/dcds.2008.21.277
[20]

Tsung-Fang Wu. Multiplicity of positive solutions for a semilinear elliptic equation in $R_+^N$ with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1675-1696. doi: 10.3934/cpaa.2010.9.1675

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (115)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy