In this paper, we utilized the Jaulent-Miodek equation which is one of important models in particle physics and engineering. The exact traveling wave solutions for this equation “involving parameters” according to two different techniques are constructed. When these parameters are taken as special values, the solitary wave solutions are derived from it. A comparison between the obtained results using these two different methods with that obtained by previous authors is demonstrated.
Matsuno, Y. (2001) Reduction of Dispersionless Coupled Korteweg-de Vries Equations to the Euler-Darboux Equation. Journal of Mathematical Physics, 42, 1744-1760. https://doi.org/10.1063/1.1345500
Sklyanin, E.K. (1995) Separation of Variables: New Trends. Progress of Theoretical Physics Supplement, 118, 35-60. https://doi.org/10.1143/PTPS.118.35
Antonowicz, M. and Rauch-Wojciechowski, S. (1992) Lax Representation for Restricted Flows of the KdV Hierarchy and for the Kepler Problem. Physics Letters A, 171, 303-310. https://doi.org/10.1016/0375-9601(92)90648-6
Jawad, A.J.M., Petkovic, M.D. and Biswas, A. (2010) Modified Simple Equation Method for Nonlinear Evolution Equations. Applied Mathematics and Computation, 217, 869-877. https://doi.org/10.1016/j.amc.2010.06.030
Kamruzzaman, K., Ali Akbar, M. and Hj Mohd Ali, N. (2013) The Modified Simple Equation Method for Exact and Solitary Wave Solutions of Nonlinear Evolution Equation: The GZK-BBM Equation and Right-Handed Non Commutative Burgers Equations. ISRN Mathematical Physics, 2013, Article ID: 146704. https://doi.org/10.1155/2013/146704
Ahmet Beki, R. and Uygun, F. (2012) Exact Travelling Wave Solutions of Some Nonlinear Evolution Equations by Using (GG)-Expansion Method. Arab Journal of Mathematical Sciences, 18, 73-85.
Zahran, E.H.M. and Khater, M.M.A. (2014) Exact Solutions to Some Nonlinear Evolution Equations by Using (G’//G)-Expansion Method. Joukull Journal, 64, 226-238.
Zhang, J.L., Wang, M.L., Wang, Y.M. and Fang, Z.D. (2006) The Improved F-Expansion Method and Its Applications. Physics Letters A, 350, 103-109. https://doi.org/10.1016/j.physleta.2005.10.099
Wang, M.L., Zhang, J.L. and Li, X.Z. (2008) The (GG)-Expansion Method and Travelling Wave Solutions of Nonlinear Evolutions Equations in Mathematical Physics. Physics Letters A, 372, 417-423. https://doi.org/10.1016/j.physleta.2007.07.051
Zahran, E.H.M. and Khater, M.M.A. (2015) The Tow Variable (G’/G, 1/G)-Expansion Method for Solving Nonlinear Dynamics of Microtubules—A New Model. Global Journal of Science Frontier Research, 15, 87-94.
Zahran, E.H.M. and Khater, M.M.A. (2014) Exact Travelling Wave Solutions for the System of Shallow Water Wave Equation Modified Liouvill Equation Using Extended Jacobi Elliptic Function Expansion Method. American Journal of Computational Mathematics, 4, 455-463. https://doi.org/10.4236/ajcm.2014.45038
Shehata, M.S.M. (2015) Extended Jacobi Elliptic Function Expansion Method and its Applications for Solving Some Nonlinear Evolution Equations in Mathematical Physics. International Journal of Computer Applications, 109, 1-4. https://doi.org/10.5120/19237-0621
Zahran, E.H.M. (2015) Exact Traveling Wave Solutions for Nano-Solitons of Ionic Waves Propagation along Microtubules in Living Cells & Nano-Ionic Currents of MTs. World Journal of Nano Science and Engineering, 5, 78-87. https://doi.org/10.4236/wjnse.2015.53010
Zahran, E.H.M. (2015) Exact Traveling Wave Solutions for Nano-Ionic Solitons & Nano-Ionic Currents of MTs Using Exp(-φ(ξ))-Expansion Method. Advances in Nano Particles, 4, 25-36. https://doi.org/10.4236/anp.2015.42004
Zahran, E.H.M. (2015) Exact Traveling Wave Solutions for Nonlinear Fractional Partial Differential Equations Arising in Soliton Using the Exp(-φ(ξ))-Expansion Method. International Journal of Computer Applications, 109, 12-17. https://doi.org/10.5120/19247-0619
Shehata, M.S.M. (2015) The Exp(-φ(ξ))-Method and Its Applications for Solving some Nonlinear Evolution Equations in Mathematical Physics. American Journal of Computational Mathematics, 5, 468-480. https://doi.org/10.4236/ajcm.2015.54041
Zahran, E.H.M. (2015) Travelling Wave Solutions of Non Linear Evolution Equation Via Modified Exp(-φ(ξ))-Expansion Method. Journal of Computational and Theoretical Nano Science, 12, 5716-5724. https://doi.org/10.1166/jctn.2015.4707
Yang, X.-F., Deng, Z.-C. and Wei, Y. (2015) A Riccati-Bernoulli Sub-ODE Method for Nonlinear Partial Differential Equations and Its Application. Advances in Difference Equations, 1, 1-17.
Shehata, M.S.M. (2016) A New Solitary Wave Solution of the Perturbed Nonlinear Schrodinger Equation Using a Riccati Bernoulli Sub-ODE Method. International Journal of Physical Sciences, 11, 80-84. https://doi.org/10.5897/IJPS2015.4442
Bekir, A. and Boz, A. (2007) Exact Solutions for a Class of Nonlinear Partial Differential Equations Using Exp-Function Method. International Journal of Nonlinear Sciences and Numerical Simulation, 8, 505-512. https://doi.org/10.1515/IJNSNS.2007.8.4.505
Fan, E.G. (2000) Extended Tanh-Function Method and Its Applications to Nonlinear Equations. Physics Letters A, 277, 212-218. https://doi.org/10.1016/S0375-9601(00)00725-8
Shehata, M.S.M. (2016) Exact Traveling Wave Solutions for Nonlinear Evolutions Equation. Journal of Computational and Theoretical Nano Science, 13, 534-538. https://doi.org/10.1166/jctn.2016.4837
Zahran, E.H.M. and Khater, M.M.A. (2016) Modified Extended Tanh-Function Method and Its Applications to the Bogoyavlenskii Equation. Applied Mathematical Modeling, 40, 1769-1775. https://doi.org/10.1016/j.apm.2015.08.018
Lu, D.C., Khater Mostafa, M.A. and Zahran, E.H.M. (2017) Solitary Wave Solutions of the Benjamin Bona-Mahoney-Burgers Equation with Dual Power-Law Nonlinearity. Applied Mathematics & Information Sciences, 11, 1-5.
Ucar, Y., Karaagas, B. and Esen, A. (2015) A New Approach on Numerical Solutions of the Improved Boussinesq Type Equation Using Quadratic B-Spline Galerkin Finite Element Method. Applied Mathematics and Computation, 270, 148-155. https://doi.org/10.1016/j.amc.2015.08.007
Jaulent, M. and Miodek, I. (1976) Nonlinear Evolution Equations Associated with “Energy-Dependent Schrödinger Potentials”. Letters in Mathematical Physics, 1, 243-250.
Wazwaz, A.-M. (2009) Multiple Kink Solutions and Multiple Singular Kink Solutions for (2+1)-Dimensional Nonlinear Models Generated by the Jaulent-Miodek Hierarchy. Physics Letters A, 373, 1844-1846. https://doi.org/10.1016/j.physleta.2009.03.049
Mohebbi, A., Asgari, Z. and Dehghan, M. (2012) Numerical Solution of Nonlinear Jaulent-Miodek and Whitham-Broer-Kaup Equations. Communications in Nonlinear Science and Numerical Simulation, 17, 4602-4610. https://doi.org/10.1016/j.cnsns.2012.04.011
Biazar, J. and Eslami, M. (2010) Homotopy Analysis Method for Nonlinear Jaulent-Miodek Equation. Journal of Information and Computing Science, 5, 83-88.
Liu, H. and Yan, F. (2011) The Bifurcation and Exact Travelling Wave Solutions for (2+1)-Dimensional Nonlinear Models Generated by the Jaulent-Miodek Hierarchy. International Journal of Nonlinear Science, 11, 200-205.