JAMP  Vol.3 No.12 , December 2015
Analytical Treatment of the Evolutionary (1 + 1)-Dimensional Combined KdV-mKdV Equation via the Novel (G'/G)-Expansion Method
The novel (G'/G)-expansion method is a powerful and simple technique for finding exact traveling wave solutions to nonlinear evolution equations (NLEEs). In this article, we study explicit exact traveling wave solutions for the (1 + 1)-dimensional combined KdV-mKdV equation by using the novel (G'/G)-expansion method. Consequently, various traveling wave solutions patterns including solitary wave solutions, periodic solutions, and kinks are detected and exhibited.

Cite this paper
Alam, M. , Belgacem, F. and Akbar, M. (2015) Analytical Treatment of the Evolutionary (1 + 1)-Dimensional Combined KdV-mKdV Equation via the Novel (G'/G)-Expansion Method. Journal of Applied Mathematics and Physics, 3, 1571-1579. doi: 10.4236/jamp.2015.312181.
[1]   Hafez, M.G., Alam, Md.N. and Akbar, M.A. (2015) Traveling Wave Solutions for Some Important Coupled Nonlinear Physical Models via the Coupled Higgs Equation and the Maccari System. Journal of King Saud University—Science, 27, 105-112.

[2]   Alam, Md.N., Hafez, M.G., Akbar, M.A. and Roshid, H.O. (2015) Exact Traveling Wave Solutions to the (3 + 1)-Dimensional mKdV-ZK and the (2 + 1)-Dimensional Burgers Equations via Exp(-Eta)-Expansion Method. Alexandria Engineering Journal, 54, 635-644.

[3]   Roshid, H.O., Alam, Md.N. and Akbar, M.A. (2015) Traveling Wave Solutions for Fifth Order (1 + 1)-Dimensional Kaup-Keperschmidt Equation with the Help of Exp(-Phi)-Expansion Method. Walailak Journal of Science and Technology, 12, 1063-1073.

[4]   Wang, M., Li, X. and Zhang, J. (2008) The (G'/G)-Expansion Method and Traveling Wave Solutions of Nonlinear Evolution Equations in Mathematical Physics. Physics Letters A, 372, 417-423.

[5]   Alam, Md.N., Akbar, M.A. and Hoque, M.F. (2014) Exact Traveling Wave Solutions of the (3 + 1)-Dimensional mKdV-ZK Equation and the (1 + 1)-Dimensional Compound KdVB Equation Using New Approach of the Generalized (G'/G)-Expansion Method. Pramana Journal of Physics, 83, 317-329.

[6]   Alam, Md.N. and Akbar, M.A. (2015) Some New Exact Traveling Wave Solutions to the Simplified MCH Equation and the (1 + 1)-Dimensional Combined KdV-mKdV Equations. Journal of the Association of Arab Universities for Basic and Applied Sciences, 17, 6-13.

[7]   Alam, Md.N. and Akbar, M.A. (2014) Traveling Wave Solutions for the mKdV Equation and the Gardner Equation by New Approach of the Generalized (G'/G)-Expansion Method. Journal of the Egyptian Mathematical Society, 22, 402-406.

[8]   Russell, J.S. (1844) Report on Waves. Proceedings of the 14th Meeting of the British Association for the Advancement of Science.

[9]   Hu, J.L. (2001) Explicit Solutions to Three Nonlinear Physical Models. Physics Letters A, 287, 81-89.

[10]   Hu, J.L. (2001) A New Method for Finding Exact Traveling Wave Solutions to Nonlinear Partial Differential Equations. Physics Letters A, 286, 175-179.

[11]   Matveev, V.B. and Salle, M.A. (1991) Darboux Transformation and Solitons. Springer, Berlin.

[12]   Salas, A.H. and Gomez, C.A. (2010) Application of the Cole-Hopf Transformation for Finding Exact Solutions to Several Forms of the Seventh-Order KdV Equation. Mathematical Problems in Engineering, 2010, Article ID: 194329.

[13]   Bock, T.L. and Kruskal, M.D. (1979) A Two-Parameter Miura Transformation of the Benjamin-Ono Equation. Physics Letters A, 74, 173-176.

[14]   Chen, Y. and Wang, Q. (2005) Extended Jacobi Elliptic Function Rational Expansion Method and Abundant Families of Jacobi Elliptic Functions Solutions to (1 + 1)-Dimensional Dispersive Long Wave Equation. Chaos, Solitons & Fractals, 24, 745-757.

[15]   Adomian, G. (1994) Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Publishers, Boston.

[16]   Wazwaz, A.M. (2002) Partial Differential Equations: Method and Applications. Taylor and Francis, London.

[17]   Li, J.B. and Liu, Z.R. (2000) Smooth and Non-Smooth Traveling Waves in a Nonlinearly Dispersive Equation. Applied Mathematical Modelling, 25, 41-56.

[18]   Liu, Z.R. and Qian, T.F. (2001) Peakons and Their Bifurcation in a Generalized Camassa-Holm Equation. International Journal of Bifurcation and Chaos, 11, 781-792.

[19]   Ablowitz, M.J. and Clarkson, P.A. (1991) Soliton, Nonlinear Evolution Equations and Inverse Scattering Method. Cambridge University Press, New York.

[20]   Malik, A., Chand, F., Kumar, H. and Mishra, S.C. (2012) Exact Solutions of the Bogoyavlenskii Equation Using the Multiple-Expansion Method. Computers & Mathematics with Applications, 64, 2850-2859.

[21]   Liao, S.J. (2005) A New Branch of Solutions of Boundary-Layer Flows over an Impermeable Stretched Plate. International Journal of Heat and Mass Transfer, 48, 2529-2539.

[22]   Liao, S.J. (2009) A General Approach to Get Series Solution of Non-Similarity Boundary-Layer Flows. Communications in Nonlinear Science and Numerical Simulation, 14, 2144-2159.

[23]   Darvishi, M.T. and Najafi, M. (2012) Some Exact Solutions of the (2 + 1)-Dimensional Breaking Soliton Equation Using the Three-Wave Method. World Academy of Science, Engineering and Technology, 87, 31-34.

[24]   Darvishi, M.T. and Najafi, M. (2012) Some Complexiton Type Solutions of the (3 + 1)-Dimensional Jimbo-Miwa Equation. World Academy of Science, Engineering and Technology, 87, 42-44.

[25]   Wang, D. and Zhang, H.Q. (2005) Further Improved F-Expansion Method and New Exact Solutions of Konopelchenko-Dubrovsky Equation. Chaos, Solitons & Fractals, 25, 601-610.

[26]   Yan, Z. (2003) Generalized Method and Its Application in the Higher-Order Nonlinear Schrodinger Equation in Nonlinear Optical Fibres. Chaos, Solitons & Fractals, 16,759-766.

[27]   Kudryashov, N.A. (1990) Exact Solutions of the Generalized Kuramoto-Sivashinsky Equation. Physics Letters A, 147, 287-291.

[28]   Alam, Md.N., Akbar, M.A. and Mohyud-Din, S.T. (2014) A Novel (G'/G)-Expansion Method and Its Application to the Boussinesq Equation. Chinese Physics B, 23, Article ID: 020203.

[29]   Alam, Md.N. and Akbar, M.A. (2014) Traveling Wave Solutions of the Nonlinear (1 + 1)-Dimensional Modified Benjamin-Bona-Mahony Equation by Using Novel (G'/G)-Expansion Method. Physical Review & Research International, 4, 147-165.

[30]   Alam, Md.N. (2015) Exact Solutions to the Foam Drainage Equation by Using the New Generalized (G'/G)-Expansion Method. Results in Physics, 5, 168-177.

[31]   Alam, Md.N., Hafez, M.G., Belgacem, F.B.M. and Akbar, M.A. (2015) Applications of the Novel (G'/G)-Expansion Method to Find New Exact Traveling Wave Solutions of the Nonlinear Coupled Higgs Field Equation. Nonlinear Studies, 22, 613-633.

[32]   Hafez, M.G., Alam, Md.N. and Akbar, M.A. (2014) Exact Traveling Wave Solutions to the Klein-Gordon Equation Using the Novel (G'/G)-Expansion Method. Results in Physics, 4, 177-184.

[33]   Alam, Md.N. and Akbar, M.A. (2015) A Novel (G'/G)-Expansion Method for Solving the (3 + 1)-Dimensional Modified KdV-Zakharov-Kuznetsov Equation in Mathematical Physics. International Journal of Computing Science and Mathematics, 6, 404-415.

[34]   Belgacem, F.B.M. and Smaoui, N. (2001) Interactions of Parabolic Convective Diffusion Equations and Navier-Stokes Equations Connected with Population Dispersal. Comm. Applied Nonlinear Anal, 8, 47-67.

[35]   Smaoui, N. and Belgacem, F.B.M. (2002) Connections between the Convective Diffusion Equation and the Forced Burgers Equation. Journal of Applied Mathematics and Stochastic Analysis, 15, 57-75.

[36]   Eckstein, E. and Belgacem, F.B.M. (1991) Model of Platelet Transport in Flowing Blood with Drift and Diffusion Terms. Biophysical Journal, 60, 53-69.

[37]   Alam, Md.N. and Belgacem, F.B.M. (2015) Application of the Novel (G'/G)-Expansion Method to the Regularized Long Wave Equation. Waves, Wavelets and Fractals—Advanced Analysis, 1, 20-36.

[38]   Zhu, S. (2008) The Generalized Riccati Equation Mapping Method in Non-Linear Evolution Equation: Application to (2 + 1)-Dimensional Boiti-Leon-Pempinelle Equation. Chaos, Solitons & Fractals, 37, 1335-1342.