AM  Vol.6 No.8 , July 2015
Application of Hyperbola Function Method to the Family of Third Order Korteweg-de Vries Equations
Author(s) Luwai Wazzan
ABSTRACT
In this work, we apply a hyperbola function method to solve the nonlinear family of third order Korteweg-de Vries equations. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions and trigonometric functions. The method used is a promising method to solve other nonlinear evaluation equations.

Cite this paper
Wazzan, L. (2015) Application of Hyperbola Function Method to the Family of Third Order Korteweg-de Vries Equations. Applied Mathematics, 6, 1241-1249. doi: 10.4236/am.2015.68117.
References
[1]   Zabusky, N.J. and Kruskal, M.D. (1965) Interaction of “Solitons” in a Collisionless Plasma and the Recurrence of Initial States. Physical Review Letters, 15, 240.
http://dx.doi.org/10.1103/PhysRevLett.15.240

[2]   Ablowitz, M.J. and Clarkson, P.A. (1992) Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge.

[3]   Hirota, R. (1973) Exact N-Soliton Solutions of the Wave Equation of Long Waves in Shallow-Water and in Nonlinear Lattices. Journal of Mathematical Physics, 14, 810.
http://dx.doi.org/10.1063/1.1666400

[4]   Wadati, M. (1975) Wave Propagation in Nonlinear Lattice. II. Journal of the Physical Society of Japan, 38, 681.
http://dx.doi.org/10.1143/JPSJ.38.681

[5]   Wang, M. and Li, X. (2005) Extended F-Expansion Method and Periodic Wave Solutions for the Generalized Zakharov Equations. Physics Letters A, 343, 48.
http://dx.doi.org/10.1016/j.physleta.2005.05.085

[6]   Fan, F. (2000) Two New Applications of the Homogeneous Balance Method. Physics Letters A, 265, 353.
http://dx.doi.org/10.1016/S0375-9601(00)00010-4

[7]   Lan, H. and Wang, K. (1990) Exact Solutions for Two Nonlinear Equations I. Journal of Physics A: Mathematical and General, 23, 3923.
http://dx.doi.org/10.1088/0305-4470/23/17/021

[8]   Yan, Z. (2003) Symbolic Computation and Weierstrass Elliptic Function Solution of The Davey—Stewartson (DS) System. International Journal of Modern Physics C, 14, 1127.
http://dx.doi.org/10.1142/S0129183103005224

[9]   Korteweg, D.J. and de Vries, G. (1895) On the Change of Form of Long Waves Advancing in a Rectangular Canal and on a New Type of Long Stationary Waves. Philosophical Magazine, 36, 422-443.
http://dx.doi.org/10.1080/14786449508620739

[10]   Schamel, H. (1973) A Modified Korteweg-de Vries Equation for Ion Acoustic Wavess Due to Resonant Electrons. Journal of Plasma Physics, 9, 377-387.
http://dx.doi.org/10.1017/S002237780000756X

[11]   Drazin, P.G. and Johnson, R.S. (1996) Solitons: An Introduction. Cambridge University Press, Cambridge.

[12]   Whitham, G.B. (1999) Linear and Nonlinear Waves. Wiley-Interscience, New York.
http://dx.doi.org/10.1002/9781118032954

[13]   Marchant, T.R. and Smyth, N.F. (1996) Soliton Interaction for the Extended Korteweg-De Vries Equation. IMA Journal of Applied Mathematics, 56, 157-176.
http://dx.doi.org/10.1093/imamat/56.2.157

[14]   Wazwaz, A.-M. (2008) Chapter 9: The KdV Equation. In: Dafermos, C.M. and Pokorny, M., Eds., Handbook of Differential Equations: Evolutionary Equations, Volume 4, Elsevier, Amsterdam, 485-568.

[15]   Malfliet, W. (1992) Solitary Wave Solutions of Nonlinear Wave Equations. American Journal of Physics, 60, 650-654.
http://dx.doi.org/10.1119/1.17120

[16]   Wazwaz, A.M. (2007) The Extended Tanh Method for New Solitons Solutions for Many Forms of the Fifth-Order KdV Equations. Applied Mathematics and Computation, 184, 1002-1014.
http://dx.doi.org/10.1016/j.amc.2006.07.002

[17]   El-Wakil, S.A., El-labany, S.K., Zahran, M.A. and Sabry, R. (2005) Modified Extended Tanh-Function Method and Its Applications to Nonlinear Equations. Applied Mathematics and Computation, 161, 403-412.
http://dx.doi.org/10.1016/j.amc.2003.12.035

[18]   El-Wakil, S.A., El-labany, S.K., Zahran, M.A. and Sabry, R. (2002) Modified Extended Tanh-Function Method for Solving Nonlinear Partial Differential Equations. Physics Letters A, 299, 179-188.
http://dx.doi.org/10.1016/S0375-9601(02)00669-2

[19]   Wazzan, L. (2009) A Modified Tanh—Coth Method for Solving the KdV and the KdV—Burgers’ Equations. Communications in Nonlinear Science and Numerical Simulation, 14, 443-450.
http://dx.doi.org/10.1016/j.cnsns.2007.06.011

[20]   Wazzan, L. (2009) A Modified Tanh—Coth Method for Solving the General Burgers—Fisher and the Kuramoto— Sivashinsky Equations. Communications in Nonlinear Science and Numerical Simulation, 14, 2642-2652.
http://dx.doi.org/10.1016/j.cnsns.2008.08.004

[21]   Wazwaz, A.M. (2002) Partial Differential Equations: Methods and Applications. Balkema Publishers, Leiden.

[22]   Ablowitz, M.J. and Clarkson, P.A. (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9780511623998

[23]   Hirota, R. (2004) The Direct Method in Soliton Theory. Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9780511543043

[24]   Hereman, W. and Zhuang, W. (1980) Symbolic Software for Soliton Theory. Acta Applicandae Mathematica, 39, 361-378.

[25]   Bai, C.-L. (2001) Exact Solutions for Nonlinear Partial Differential Equation: A New Approach. Physics Letters A, 288, 191-195.
http://dx.doi.org/10.1016/S0375-9601(01)00522-9

 
 
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