The exp(-φ(ξ)) -expansion method is used as the first time to investigate the wave solution of a nonlinear dynamical system in a new double-Chain model of DNA and a diffusive predator-prey system. The proposed method also can be used for many other nonlinear evolution equations.
KeywordsDynamical System in a New Double-Chain Model of DNAA Diffusive Predator-Prey SystemTraveling Wave SolutionsSolitary Wave SolutionsKink-Anti Kink Shaped
Ablowitz, M.J. and Segur, H. (1981) Solitions and Inverse Scattering Transform. SIAM, Philadelphia.
Zahran, E.H.M. and Khater, M.M.A. (2014) Exact Traveling Wave Solutions for the System of Shallow Water Wave Equations and Modified Liouville Equation Using Extended Jacobian Elliptic Function Expansion Method. American Journal of Computational Mathematics (AJCM), 4.
Zahran, E.H.M. and Khater, M.M.A. (2014) The Modified Simple Equation Method and Its Applications for Solving Some Nonlinear Evolutions Equations in Mathematical Physics. Jokull Journal, 64.
Wazwaz, A.M. (2004) The Tanh Method for Travelling Wave Solutions of Nonlinear Equations. Applied Mathematics and Computation, 154, 714-723. http://dx.doi.org/10.1016/S0096-3003(03)00745-8
EL-Wakil, S.A. and Abdou, M.A. (2007) New Exact Travelling Wave Solutions Using Modified Extented Tanh-Function Method. Chaos Solitons Fractals, 31, 840-852.
Fan, E. (2000) Extended Tanh-Function Method and Its Applications to Nonlinear Equations. Physics Letters A, 277, 212-218.
Abdelrahman, M.A.E., Zahran E.H.M. and Khater, M.M.A. (2015) Exact Traveling Wave Solutions for Modified Liouville Equation Arising in Mathematical Physics and Biology. International Journal of Computer Applications (0975 8887), 112.
Wazwaz, A.M. (2005) Exact Solutions to the Double Sinh-Gordon Equation by the Tanh Method and a Variable Separated ODE Method. Computers & Mathematics with Applications, 50, 1685-1696. http://dx.doi.org/10.1016/j.camwa.2005.05.010
Wazwaz, A.M. (2004) A Sine-Cosine Method for Handling Nonlinear Wave Equations. Mathematical and Computer Modelling, 40, 499-508. http://dx.doi.org/10.1016/j.mcm.2003.12.010
Yan, C. (1996) A Simple Transformation for Nonlinear Waves. Physics Letters A, 224, 77-84. http://dx.doi.org/10.1016/S0375-9601(96)00770-0
Fan, E. and Zhang, H. (1998) A Note on the Homogeneous Balance Method. Physics Letters A, 246, 403-406. http://dx.doi.org/10.1016/S0375-9601(98)00547-7
Wang, M.L. (1996) Exact Solutions for a Compound KdV-Burgers Equation. Physics Letters A, 213, 279-287. http://dx.doi.org/10.1016/0375-9601(96)00103-X
Abdou, M.A. (2007) The Extended F-Expansion Method and Its Application for a Class of Nonlinear Evolution Equations. Chaos, Solitons & Fractals, 31, 95-104. http://dx.doi.org/10.1016/j.chaos.2005.09.030
Ren, Y.J. and Zhang, H.Q. (2006) A Generalized F-Expansion Method to Find Abundant Families of Jacobi Elliptic Function Solutions of the (2+1)-Dimensional Nizhnik-Novikov-Veselov Equation. Chaos, Solitons & Fractals, 27, 959-979. http://dx.doi.org/10.1016/j.chaos.2005.04.063
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. http://dx.doi.org/10.1016/j.physleta.2005.10.099
He, J.H. and Wu, X.H. (2006) Exp-Function Method for Nonlinear Wave Equations. Chaos, Solitons & Fractals, 30, 700-708. http://dx.doi.org/10.1016/j.chaos.2006.03.020
Aminikhad, H., Moosaei, H. and Hajipour, M. (2009) Exact Solutions for Nonlinear Partial Differential Equations via Exp-Function Method. Numerical Methods for Partial Differential Equations, 26, 1427-1433.
Zhang, Z.Y. (2008) New Exact Traveling Wave Solutions for the Nonlinear Klein-Gordon Equation. Turkish Journal of Physics, 32, 235-240.
Wang, M.L., Zhang, J.L. and Li, X.Z. (2008) The -Expansion Method and Travelling Wave Solutions of Nonlinear Evolutions Equations in Mathematical Physics. Physics Letters A, 372, 417-423. http://dx.doi.org/10.1016/j.physleta.2007.07.051
Zhang, S., Tong, J.L. and Wang, W. (2008) A Generalized -Expansion Method for the mKdv Equation with Variable Coefficients. Physics Letters A, 372, 2254-2257. http://dx.doi.org/10.1016/j.physleta.2007.11.026
Abdelrahman, M.A.E., Zahran, E.H.M. and Khater, M.M.A. (2015) Exact Traveling Wave Solutions for Modified Liouville Equation Arising in Mathematical Physics and Biology. International Journal of Computer Applications (0975 8887), 112.
Zahran, E.H.M. and Khater, M.M.A. (2014) Exact Solutions to Some Nonlinear Evolution Equations by Using (G’/G)-Expansion Method. Jokull Journal, 64.
Dai, C.Q. and Zhang, J.F. (2006) Jacobian Elliptic Function Method for Nonlinear Differential Difference Equations. Chaos, Solitons & Fractals, 27, 1042-1049. http://dx.doi.org/10.1016/j.chaos.2005.04.071
Fan, E. and Zhang, J. (2002) Applications of the Jacobi Elliptic Function Method to Special-Type Nonlinear Equations. Physics Letters A, 305, 383-392. http://dx.doi.org/10.1016/S0375-9601(02)01516-5
Liu, S., Fu, Z., Liu, S. and Zhao, Q. (2001) Jacobi Elliptic Function Expansion Method and Periodic Wave Solutions of Nonlinear Wave Equations. Physics Letters A, 289, 69-74. http://dx.doi.org/10.1016/S0375-9601(01)00580-1
Zhao, X.Q., Zhi, H.Y., Zhang, H.Q. (2006) Improved Jacobi-Function Method with Symbolic Computation to Construct New Double-Periodic Solutions for the Generalized Ito System. Chaos Solitons Fractals, 28, 112-126.
Abdelrahman, M.A.E., Zahran, E.H. M. and Khater, M.M.A. (2014) Exact Traveling Wave Solutions for Power Law and Kerr Law Non Linearity Using the -Expansion Method. GJSFR, Volume 14, Issue 4, Version 1.0.
Mahmoud A.E. Abdelrahman and Mostafa M.A. Khater (2015) The Expansion Method and Its Application for Solving Nonlinear Evolution Equations. International Journal of Science and Research (IJSR), 4, 2319-7064.
Khan, K. and Akbar, M.A. (2014) Solitary Wave Solutions of Some Coupled Nonlinear Evolution Equations. Journal of Scientific Research, 6, 273-284
Aguero, M., Najera, M. and Carrillo, M. (2008) Non Classic Solitonic Structures in DNA’s Vibrational Dynamics. International Journal of Modern Physics B, 22, 2571-2582. http://dx.doi.org/10.1142/S021797920803968X
Gaeta, G. (1999) Results and Limitations of the Soliton Theory of DNA Transcription. Journal of Biological Physics, 24, 81-96. http://dx.doi.org/10.1023/A:1005158503806
Gaeta, G., Reiss, C., Peyrard, M. and Dauxois, T. (1994) Simple Models of Nonlinear DNA Dynamics. La Rivista Del Nuovo Cimento Series 3, 17, 1-48. http://dx.doi.org/10.1007/BF02724511
Yakushevich, L.V. (1987) Nonlinear Physics of DNA. Studio Biophys, 121, 201.
Yakushevich, L.V. (1989) Nonlinear DNA Dynamics: A New Model. Physics Letters A, 136, 413-417. http://dx.doi.org/10.1016/0375-9601(89)90425-8
Yakushevich, L.V. (1998) Nonlinear Physics of DNA. Wiley and Sons, England.
Peyrard, M. and Bishop, A. (1989) Statistical Mechanics of a Nonlinear Model of DNA Denaturation. Physical Review Letters, 62, 2755-2758. http://dx.doi.org/10.1103/PhysRevLett.62.2755
Kong, D.X., Lou, S.Y. and Zeng, J. (2001) Nonlinear Dynamics in a New Double Chain-Model of DNA. Communications in Theoretical Physics, 36, 737-742. http://dx.doi.org/10.1088/0253-6102/36/6/737
Alka, W., Goyal, A. and Kumar, C.N. (2011) Nonlinear Dynamics of DNA-Riccati Generalized Solitary Wave Solutions. Physics Letters A, 375, 480-483. http://dx.doi.org/10.1016/j.physleta.2010.11.017
Petrovskii, S.V., Malchow, H. and Li, B.L. (2005) An Exact Solution of a Diffusive Predator-Prey System. Proceedings of the Royal Society A, 461, 1029-1053.
Kraenkel, R.A., Manikandan, K. and Senthivelan, M. (2013) On Certain New Exact Solutions of a Diffusive Predator-Prey System. Communications in Nonlinear Science and Numerical Simulation, 18, 1269-1274. http://dx.doi.org/10.1016/j.cnsns.2012.09.019
Dehghan, M. and Sabouri, M. (2013) A Legendre Spectral Element Method on a Large Spatial Domain to Solve the Predator-Prey System Modeling Interaction Populations. Applied Mathematical Modeling, 37, 1028-1038. http://dx.doi.org/10.1016/j.apm.2012.03.030