Exact travelling wave solutions for isothermal magnetostatic atmospheres by Fan subequation method. (English) Zbl 1259.35055

Summary: The equations of magnetohydrostatic equilibria for a plasma in a gravitational field are investigated analytically. An investigation of a family of isothermal magnetostatic atmospheres with one ignorable coordinate corresponding to a uniform gravitational field in a plane geometry is carried out. These equations transform to a single nonlinear elliptic equation for the magnetic vector potential \(u\). This equation depends on an arbitrary function of \(u\) that must be specified. With choices of the different arbitrary functions, we obtain analytical solutions of elliptic equation using the Fan subequation method.


35C07 Traveling wave solutions
35Q35 PDEs in connection with fluid mechanics
35Q60 PDEs in connection with optics and electromagnetic theory
35J60 Nonlinear elliptic equations
Full Text: DOI


[1] A. H. Khater, M. A. El-Attary, M. F. El-Sabbagh, and D. K. Callebaut, “Two-dimensional magnetohydrodynamic equilibria,” Astrophysics and Space Science, vol. 149, no. 2, pp. 217-223, 1988. · Zbl 0662.76139
[2] A. H. Khater, D. K. Callebaut, and O. H. El-Kalaawy, “Bäcklund transformations and exact solutions for a nonlinear elliptic equation modelling isothermal magnetostatic atmosphere,” IMA Journal of Applied Mathematics, vol. 65, no. 1, pp. 97-108, 2000. · Zbl 0959.35056
[3] I. Lerche and B. C. Low, “Some nonlinear problems in astrophysics,” Physica D. Nonlinear Phenomena, vol. 4, no. 3, pp. 293-318, 1981/82. · Zbl 1194.37169
[4] B. C. Low, “Evolving force-free magnetic elds. I-the development of the pre are stage,” The Astrophysical Journal, vol. 212, pp. 234-242, 1977.
[5] X.-H. Wu and J.-H. He, “Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 966-986, 2007. · Zbl 1143.35360
[6] W. Zwingmann, “Theoretical study of onset conditions for solar eruptive processes,” Solar Physics, vol. 111, pp. 309-331, 1987.
[7] I. Lerche and B. C. Low, “On the equilibrium of a cylindrical plasma supported horizontally by magneticelds in uniform gravity,” Solar Physics, vol. 67, pp. 229-243, 1980.
[8] G. M. Webb, “Isothermal magnetostatic atmospheres. II-similarity solutions with current proportional to the magnetic potential cubed,” The Astrophysical Journal, vol. 327, pp. 933-949, 1988.
[9] G. M. Webb and G. P. Zank, “Application of the sine-Poisson equation in solar magnetostatics,” Solar Physics, vol. 127, pp. 229-252, 1990.
[10] E. G. Fan and Y. C. Hon, “A series of travelling wave solutions for two variant Boussinesq equations in shallow water waves,” Chaos, Solitons & Fractals, vol. 15, no. 3, pp. 559-566, 2003. · Zbl 1031.76008
[11] D. Feng and K. Li, “Exact traveling wave solutions for a generalized Hirota-Satsuma coupled KdV equation by Fan sub-equation method,” Physics Letters A, vol. 375, no. 23, pp. 2201-2210, 2011. · Zbl 1241.35178
[12] S. A. El-Wakil and M. A. Abdou, “The extended Fan sub-equation method and its applications for a class of nonlinear evolution equations,” Chaos, Solitons and Fractals, vol. 36, no. 2, pp. 343-353, 2008. · Zbl 1350.35170
[13] E. Yomba, “The modified extended Fan sub-equation method and its application to the (2+1)-dimensional Broer-Kaup-Kupershmidt equation,” Chaos, Solitons and Fractals, vol. 27, no. 1, pp. 187-196, 2006. · Zbl 1088.35532
[14] E. Yomba, “The extended Fan’s sub-equation method and its application to KdV-MKdV, BKK and variant Boussinesq equations,” Physics Letters A, vol. 336, no. 6, pp. 463-476, 2005. · Zbl 1136.35451
[15] S. Zhang and H.-Q. Zhang, “Fan sub-equation method for Wick-type stochastic partial differential equations,” Physics Letters A, vol. 374, no. 41, pp. 4180-4187, 2010. · Zbl 1238.35198
[16] S. Zhang and T. Xia, “A further improved extended Fan sub-equation method and its application to the (3+1)-dimensional Kadomstev-Petviashvili equation,” Physics Letters A, vol. 356, no. 2, pp. 119-123, 2006. · Zbl 1160.37404
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.