Traveling Wave Solutions for Fifth Order (1+1)-Dimensional Kaup-Keperschmidt Equation with the Help of Exp(-Phi)-Expansion Method
By using exp(-Phi)-expansion method, abundant exact traveling wave solutions for the fifth order (1+1)-dimensional Kaup-Keperschmidt equation have been obtained in a uniform way. The obtained solutions in this work are imperative and significant for the explanation of some practical physical phenomena. It is shown that the exp(-Phi)-expansion method together with the first order ordinary differential equation, provides a progress mathematical tool for solving nonlinear partial differential equations. Numerical results, together with graphical representation, explicitly reveal the complete reliability and high efficiency of the proposed algorithm.
SJ Liao. A new branch of solutions of boundary-layer flows over an impermeable stretched plate, Int. J. Heat Mass Tran. 2005; 48, 2529-39.
SJ Liao. A general approach to get series solution of non-similarity boundary-layer flows. Commun. Nonlinear Sci. Numer. Simulat. 2009; 14, 2144-59.
MT Darvishi and M Najafi. Some exact solutions of the (2+1)-dimensional breaking soliton equation using the three-wave method. World Acad. Sci. Eng. Tech. 2012; 87, 31-4.
T Darvishi and M Najafi. Some complexiton type solutions of the (3+1)-dimensional Jimbo-Miwa equation. World Acad. Sci. Eng. Tech. 2012; 87, 42-4.
D Wang and HQ Zhang. Further improved F-expansion method and new exact solutions of Konopelchenko-Dubrovsky equation. Chaos Soliton. Fract. 2005; 25, 601-10.
Z Yan. Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres. Chaos Soliton. Fract. 2003; 16, 759-66.
NA Kudryashov. Exact solutions of the generalized Kuramoto-Sivashinsky equation. Phys. Lett. A 1990; 147, 287-91.
Y Chen and Q Wang. Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic functions solutions to (1+1)-dimensional dispersive long wave equation. Chaos Soliton. Fract. 2005; 24, 745-57.
S Liu, Z Fu, SD Liu and Q Zhao. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A 2001; 289, 69-74.
W Malfliet. The tanh method: A tool for solving certain classes of nonlinear evolution and wave equations. J. Comput. Appl. Math. 2004; 164-165, 529-41.
W Malfliet. Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 1992; 60, 650-4.
MA Abdou. The extended tanh-method and its applications for solving nonlinear physical models. Appl. Math. Comput. 2007; 190, 988-96.
AM Wazwaz. The extended tanh-method for new compact and non-compact solutions for the KP-BBM and the ZK-BBM equations. Chaos Soliton. Fract. 2008; 38, 1505-16.
MJ Ablowitz and PA Clarkson. Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform. Cambridge University Press, Cam-Bridge, 1991.
R Hirota. Exact solution of the KdV equation for multiple collisions of solutions. Phys. Rev. Lett. 1971; 27, 1192-4.
C Rogers and WF Shadwick. Backlund Transformations and Their Applications. Academic Press, New York, 1984.
JH He and XH Wu. Exp-function method for nonlinear wave equations. Chaos Soliton. Fract. 2006; 30, 700-8.
H Naher, FA Abdullah and MA Akbar. New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the Exp-function method. J. Appl. Math. 2012; 2012, 575387.
ST Mohyud-Din, MA Noor and A Waheed. Exp-function method for generalized travelling solutions of Calogero-Degasperis-Fokas equation. J. Phys. Sci. 2010; 65, 78-84.
MA Akbar and NHM Ali. New solitary and periodic solutions of nonlinear volution equation by exp-function method. World Appl. Sci. J. 2012; 17, 1603-10.
NA Kudryashov. On types of nonlinear non-integrable equations with exact solutions. Phys. Lett. A 1991; 155, 269-75.
MA Abdou and AA Soliman. Modified extended tanh-function method and its application on nonlinear physical equations. Phys. Lett. A 2006; 353, 487-92.
SA El-Wakil and MA Abdou. New exact travelling wave solutions using modified extended tanh-function method. Chaos Soliton. Fract. 2007; 31, 840-52.
X Zhao, L Wang and W Sun. The repeated homogeneous balance method and its applications to nonlinear partial differential equations. Chaos Soliton. Fract. 2006; 28, 448-53.
F Zhaosheng. Comment on “On the extended applications of homogeneous balance method”. Appl. Math. Comput. 2004; 158, 593-6.
X Zhao and D Tang. A new note on a homogeneous balance method. Phys. Lett. A 2002; 297, 59-67.
DS Wang, X Zeng and YQ Ma. Exact vortex solitons in a quasi-two-dimensional Bose-Einstein condensate with spatially inhomogeneous cubic-quintic nonlinearity. Phys. Lett. A 2012; 376, 3067-70.
DS Wang, DJ Zhang and J Yang. Integrable properties of the general coupled nonlinear Schrödinger equations. J. Math. Phys. 2010; 51, 023510.
DS Wang, XH Hu, J Hu and WM Liu. Quantized quasi-two-dimensional Bose-Einstein condensates with spatially modulated nonlinearity. Phys. Rev. A 2010; 81, 025604.
DS Wang and H Li. Single and multi-solitary wave solutions to a class of nonlinear evolution equations. J. Math. Anal. Appl. 2008; 343, 273-98.
HO Roshid, N Rahman and MA Akbar. Traveling waves solutions of nonlinear Klein Gordon equation by extended (G′/G)-expansion method. Ann. Pure Appl. Math. 2013; 3, 10-6.
HO Roshid, MN Alam, MF Hoque and MA Akbar. A new extended (G′/G)-expansion method to find exact traveling wave solutions of nonlinear evolution equations. Math. Stat. 2013; 1, 162-6.
MN Alam, MA Akbar and HO Roshid. Study of nonlinear evolution equations to construct traveling wave solutions via the new approach of generalized (G′/G)-expansion method. Math. Stat. 2013; 1, 102-12.
MM Zhao and C Li. The -expansion method applied to nonlinear evolution equations. Sciencepaper Online 2008; 2008, 21789.
JM Yuan and J Wu. A dual-Petrov-Galerkin method for two integrable fifth-order KdV type equations. Discret. Contin. Dyn. Syst. 2010; 26, 1525-36.
D Kaup. On the inverse scattering problem for the cubic eigenvalue problems of the class . Stud. Appl. Math. 1989; 62, 189-216.
M Jimbo and T Miwa. Solitons and infinite dimensional Lie algebras. Publ. RIMS, Kyoto Univ. 1983; 19, 943-1001.
J Satsuma and DJ Kaup. A Bäcklund transformation for a higher order orteweg-de Vries equation. J. Phys. Soc. Jpn. 1977; 43, 692-7.
AH Salas, CA Gómez and JE Castillo. Symbolic computation of solutions for the general fifth-order KdV equation. Int. J. Nonlinear Sci. 2010; 9, 1-8.
H Goodarzian, E Ekrami and A Azadi. Application of Exp-function method for non-linear evolution equations to the periodic and soliton solutions. Indian J. Sci. Tech. 2011; 4, 85-90.
D Feng and K Li. On exact traveling wave solutions for (1+1)-dimensional Kaup-Kupershmidt equation. Appl. Math. 2011; 2, 752-6.
M Shakeel and ST Mohyud-Din. An alternative (G′/G)-expansion method with generalized Riccati equation: application to fifth order (1+1)-dimensional Kaup-Keperschmidt equation. Open J. Math. Model. 2013; 1, 173-83.