In this work, the extended Jacobian elliptic function expansion method is used as the first time to evaluate the exact traveling wave solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to nano-solitons of ionic waves propagation along microtubules in living cells and nano-ionic currents of MTs which play an important role in biology.
References
[1]
Ablowitz, M.J. and Segur, H. (1981) Solitions and Inverse Scattering Transform. SIAM, Philadelphia. http://dx.doi.org/10.1137/1.9781611970883
[2]
Maliet, W. (1992) Solitary Wave Solutions of Nonlinear Wave Equation. American Journal of Physics, 60, 650-654. http://dx.doi.org/10.1119/1.17120
[3]
Maliet, W. and Hereman, W. (1996) The Tanh Method: Exact Solutions of Nonlinear Evolution and Wave Equations. Physica Scripta, 54, 563-568. http://dx.doi.org/10.1088/0031-8949/54/6/003
[4]
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
[5]
El-Wakil, S.A. and Abdou, M.A. (2007) New Exact Travelling Wave Solutions Using Modified Extended Tanh-Function Method. Chaos Solitons Fractals, 31, 840-852. http://dx.doi.org/10.1016/j.chaos.2005.10.032
[6]
Fan, E. (2000) Extended Tanh-Function Method and Its Applications to Nonlinear Equations. Physics Letters A, 277, 212-218. http://dx.doi.org/10.1016/S0375-9601(00)00725-8
[7]
Wazwaz, A.M. (2007) The Extended Tanh Method for Abundant Solitary Wave Solutions of Nonlinear Wave Equations. Applied Mathematics and Computation, 187, 1131-1142. http://dx.doi.org/10.1016/j.amc.2006.09.013
[8]
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
[9]
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
[10]
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
[11]
Fan, E.G. and Zhang, H.Q. (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
[12]
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
[13]
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
[14]
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
[15]
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
[16]
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
[17]
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.
[18]
Zhang, Z.Y. (2008) New Exact Traveling Wave Solutions for the Nonlinear Klein-Gordon Equation. Turkish Journal of Physics, 32, 235-240.
[19]
Wang, M.L., Zhang, J.L. and Li, X.Z. (2008) The (G’/G)-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
[20]
Zhang, S., Tong, J.L. and Wang, W. (2008) A Generalized (G’/G)-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
[21]
Zayed, E.M.E. and Gepreel, K.A. (2009) The (G’/G)-Expansion Method for Finding Traveling Wave Solutions of Nonlinear Partial Differential Equations in Mathematical Physics. Journal of Mathematical Physics, 50, Article ID: 013502. http://dx.doi.org/10.1063/1.3033750
[22]
Zayed, E.M.E. (2009) The (G’/G)-Expansion Method and Its Applications to Some Nonlinear Evolution Equations in Mathematical Physics. Journal of Applied Mathematics and Computing, 30, 89-103. http://dx.doi.org/10.1007/s12190-008-0159-8
[23]
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
[24]
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
[25]
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
[26]
Zhao, X.Q., Zhi, H.Y. and 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. http://dx.doi.org/10.1016/j.chaos.2005.05.016
[27]
Sataric, M., Dragic, M. and Sejulic, D. (2011) From Giant Ocean Solitons to Cellular Ionic Nano-Solitons. Romanian Reports in Physics, 63, 624-640.
