## Journal of Applied Mathematics

### Further Results about Traveling Wave Exact Solutions of the Drinfeld-Sokolov Equations

#### Abstract

We employ the complex method to obtain all meromorphic exact solutions of complex Drinfeld-Sokolov equations (DS system of equations). The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all constant and simply periodic traveling wave exact solutions of the equations (DS) are solitary wave solutions, the complex method is simpler than other methods and there exist simply periodic solutions ${v}_{s,3}\left(z\right)$ which are not only new but also not degenerated successively by the elliptic function solutions. We believe that this method should play an important role for finding exact solutions in the mathematical physics. For these new traveling wave solutions, we give some computer simulations to illustrate our main results.

#### Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 523732, 6 pages.

Dates
First available in Project Euclid: 14 March 2014

https://projecteuclid.org/euclid.jam/1394808165

Digital Object Identifier
doi:10.1155/2013/523732

Mathematical Reviews number (MathSciNet)
MR3122105

Zentralblatt MATH identifier
06950728

#### Citation

Zhang, Fu; Qi, Jian-ming; Yuan, Wen-jun. Further Results about Traveling Wave Exact Solutions of the Drinfeld-Sokolov Equations. J. Appl. Math. 2013 (2013), Article ID 523732, 6 pages. doi:10.1155/2013/523732. https://projecteuclid.org/euclid.jam/1394808165

#### References

• Ü. Göktaş and W. Hereman, “Symbolic computation of conserved densities for systems of nonlinear evolution equations,” Journal of Symbolic Computation, vol. 24, no. 5, pp. 591–621, 1997.
• P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2nd edition, 1993.
• J. P. Wang, “A list of $1+1$ dimensional integrable equations and their properties,” Journal of Nonlinear Mathematical Physics, vol. 9, no. suppl. 1, pp. 213–233, 2002.
• A.-M. Wazwaz, “Exact and explicit travelling wave solutions for the nonlinear Drinfeld-Sokolov system,” Communications in Nonlinear Science and Numerical Simulation, vol. 11, no. 3, pp. 311–325, 2006.
• S. A. El-Wakil and M. A. Abdou, “Modified extended tanh-function method for solving nonlinear partial differential equations,” Chaos, Solitons & Fractals, vol. 31, no. 5, pp. 1256–1264, 2007.
• W. J. Yuan, Y. Z. Li, and J. M. Lin, “Meromorphic solutions of an auxiliary ordinary differential equation using complex method,” Mathematical Methods in the Applied Sciences, vol. 36, no. 13, pp. 1776–1782, 2013.
• W. Yuan, Y. Huang, and Y. Shang, “All traveling wave exact solutions of two nonlinear physical models,” Applied Mathematics and Computation, vol. 219, no. 11, pp. 6212–6223, 2013.
• W. J. Yuan, Y. D. Shang, Y. Huang, and H. Wang, “The representation of meromorphic solutions of certain ordinary differential equations and its applications,” Scientia Sinica Mathematica, vol. 43, no. 7, 2013.
• S. Lang, Elliptic Functions, vol. 112 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2nd edition, 1987.
• R. Conte and M. Musette, “Elliptic general analytic solutions,” Studies in Applied Mathematics, vol. 123, no. 1, pp. 63–81, 2009.