Exp-Expansion Method and Exact Solutions of  Two Variable Coefficients Nonlinear Evolution Equations

Abstract

Abstract: The Exp-expansion method is used successfully to solve a generalized variable coef?cients KdV–mKdV equation and (2+1)–dimensional Broer-Kaup equation.The singular travelling wave solutions are expressed in terms of the exponential functions,the hyperbolic functions,the trigonometric functions and the rational functions.When parameters are taken to be special values,the kink type solitary wave solutions are derived. It is shown that the Exp(-?(?))-expansion method is not only appropriate for solving nonlinear evolution equations with constant coefficients but also applicable to solve the nonlinear evolution equations with variable coefficients.

Key words: nonlinear evolution equation with variable coefficients; exact solutions; -expansion method; fibre-optical communication

CLC Number:

O175.29

Wang Xiao-li, Sirendaoerji. Exp-Expansion Method and Exact Solutions of Two Variable Coefficients Nonlinear Evolution Equations[J]. J4, 2016, 33(6): 680-688.

References

[1] Li Zhibin. Traveling Wave Solution of Nonlinear Mathematical and Physical Equations(非线性数学物理方法的行波解) [M]. Beijing: Science Press, 2007(in Chinese).
[2] Gu C H,Hu H S,Zhou Z X. Darboux Transformation in Soliton Theory and its Geometric Applications(孤立子理论中的Darboux变换及其几何应用)[M].Shanghai:Scientific and Technical Press,1999(in Chinese).
[3] Hirota A R. Exact solution of the KdV equation for multiple collisions of solitions[J]. Phys. Rev,Lett,1971,27:1192-1194.
[4] Steeb W H, Euler N. Nonlinear evolution equations and Painlevé test[J]. World Scientific Publishing Company, Singapore,1988.
[5] Ablowitz M J,Clarkson P A. Soliton ,Nonlinear Evolution Equations and Inverse Scattering [M]. Cambridge Univ.Press,1991:123-136.
[6] Sirendaoerji.Auxiliary equation method and solitary wave solutions to nonlinear evolution equations[J].Journal of Inner Mongolia Normal University(Natural Science Edition)(内蒙古师范大学学报（自然科学报）),2003,35(2):127-131(in Chinese).
[7] Parkes EJ,Duffy BR. An automated tanh-function for finding solitary wave solutions to non-linear evolution equations[J].Comput Phys Commun ,1996,98:288-300.
[8] Liu S K, Fu Z T, Liu S D.Expansion method about the jacobi elliptic function andits applications to nonlinear wave equationsjacobi[J]. Acta.Phys.Sin(物理学报),2001,50(11): 2068-2073(in Chinese).
[9] Wang M L,Zhang J L,Li X Z. The -expansion method and traveling wave solutions of nonlinear evolution equation in mathematical physics[J].Phys. Lett. A ,2008,372(4): 417-423.
[10] Abdelrahman, M.A.E, Zahran, E.H.M. and Khater, M.M.A. Exact Traveling Wave Solutions for Power Law and Kerr Law Non Linearity Using the Exp( ) -Expansion Method[J]. Global Journal of Science Frontier Research,2014,14:52-60.
[11] Maha S.M.Shehata. The Exp( ) Method and Its Applications for Solving Some Nonlinear Evolution Equations in Mathematical Physics[J].American Journal of Computational Mathematics, 2015,5: 468-480.
[12] Gao Q M, Yu X, Gao Y T,et al. Painlevé analysis, Lax pair, B?cklund transformation and multi-soliton solutions for a generalized variable-coef?cient KdV–mKdV equation in ?uids and plasmas[J], Phys.Scr,
2012,85:1-12.
[13] Ma S H,Ren Q B,Fang J P,et al. Special soliton structures and the phenomena of fission and
annihilation of solitons for the (2 +1)-dimensional Broer-Kaup system with variable coefficients[J]. Acta.Phys.Sin(物理学报), 2007, 56(12): 6777-6783(in Chinese).
[14] Hong B J,Lu D C. New soliton-like solutions to the (2 +1)-dimensional Broer-Kaup equation with variable coefficients[J].Journal of Atomic and Molecular Physics(分子与原子物理学报),2008, 25(1): 130-134(in Chinese).
[15] Taogetusang,Sirendaoerji. New solutions of the combined KdV equation with the variable coefficients[J].Chinese Journal of Quantum Electronics(量子电子学报),2009,26(2):148-154(in Chinese).

Exp-Expansion Method and Exact Solutions of Two Variable Coefficients Nonlinear Evolution Equations

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 10

Recommended Articles

Metrics

Comments

[1]	. Painlevé analysis and solutions of the variable coefficient Schr?dinger equation [J]. J4, 2017, 34(6): 682-690.
[2]	. The first integral and new infinite sequence solutions of (n+1)-dimensional multiple sine-Gordon equation [J]. J4, 2017, 34(3): 316-326.
[3]	WANG Xiao-Min, SUDAO Bilige. Potential symmetries and invariant solutions of two nonlinear equations [J]. J4, 2016, 33(5): 537-544.
[4]	. Finite symmetry group solutions to a generalized variable coefficients (3+1)-dimensional nonlinear Schr\"{o}ding equation [J]. J4, 2016, 33(3): 263-278.
[5]	Li Ning, Taogetusang. New infinite sequence soliton-like solutions of the (2+1)-dimension general Calogero-Bogoyavlenskii-Schiff system [J]. J4, 2015, 32(4): 414-418.
[6]	FENG Chun-ming，LIU Qing-song. Symmetries, reductions, exact solutions and conservation laws of the Zabolotskaya-KhokhlovEquation with a Dissipative Term [J]. J4, 2015, 32(1): 46-52.
[7]	MENG Xiang-hua, XU Rui-lin, XU Xiao-ge. Auto-Backlund transformation and novel exact analytic solutions for the generalized variable-coefficient Burgers-Kadomtsev-Petviashvili equation [J]. J4, 2014, 31(6): 663-669.
[8]	Aruna，Taogetusang. New infinite sequence solutions for simplified model of GP equation in the 1D-Tonks-Girardeau gas [J]. J4, 2014, 31(5): 541-546.
[9]	PANG Jing, JIN Ling-hua，YING Xiao-mei. Solving generalized Burgers equation with variable coefficients by (G'/G) -expansion method [J]. J4, 2011, 28(6): 674-681.
[10]	TAOGeTuSang, SIRenDaoErJi. Jacobi elliptic function exact solutions of sine-Gordon equation [J]. J4, 2009, 26(3): 278-287.