Lie symmetry analysis and traveling wave solutions of equal width wave equation
Keywords:Equal width wave (EWW) equation, Lie symmetry analysis method, Power series solutions, Tanh method, Symbolic computation
AbstractWe obtained the power series solution and the traveling wave solutions of equal width wave equation by using the Lie symmetry method. The fundamental idea behind the symmetry transformation method is that it reduces one independent variables in a system of PDEs by utilizing Lie symmetries and surface invariance condition. We first obtained the infinitesimals and commutation table with the help of MAPLE software. Lie symmetry transformation method (STM) has been applied on EWW equation and converted it into various nonlinear ODEs. Then, the tanh method and the power series method have been applied for solving the reduced nonlinear ordinary differential equations (ODEs). Convergence of the power series solutions has also been shown.
M. Kaplan, A. Akbulut, and A. Bekir, “Exact travelling wave solutions of the nonlinear evolution equations by auxiliary equation method”, Zeitschrift für naturforschung A, vol. 70, no. 11, pp. 969–974, Oct. 2015, doi: 10.1515/zna-2015-0122.
P. J. Morrison, J. D. Meiss, and J. R. Cary, “Scattering of regularized-long-wave solitary waves”, Physica D: nonlinear phenomena, vol. 11, no. 3, pp. 324-336, Jun. 1984, doi: 10.1016/0167-2789(84)90014-9.
J-H. He, “Variational iteration methoda kind of non-linear analytical technique: some examples”, International journal of non-linear mechanics, vol. 34, no. 4, pp. 699-708, Jul. 1999, doi: 10.1016/S0020-7462(98)00048-1.
E. Yusufoglu and A. Bekir, “Numerical solution of equal-width wave equation”, Computers & mathematics with applications, vol. 54, no. 7-8, pp. 1147-1153, Oct. 2007, doi: 10.1016/j.camwa.2006.12.080.
R.-J. Cheng and H.-X. Ge, “Analysis of the equal width wave equation with the mesh-free reproducing kernel particle Ritz method”, Chinese physics B, vol. 21, no. 10, Art. ID. 100209, Oct. 2012, doi: 10.1088/1674-1056/21/10/100209.
A. Bihlo and R.O. Popovych, “Group classification of linear evolution equations”, Journal of mathematical analysis and applications, vol. 448, no. 2, pp. 982-1005, Apr. 2017, doi: 10.1016/j.jmaa.2016.11.020.
N. C. Roşca, A. V. Roşca, and I. Pop, “Lie group symmetry method for MHD double-diffusive convection from a permeable vertical stretching/shrinking sheet”, Computers & mathematics with applications, vol. 71, no. 8, pp. 1679-1693, Apr. 2016, doi: 10.1016/j.camwa.2016.03.006.
Z. Y. Ma, H. L. Wu, and Q. Y. Zhu, “Lie symmetry, full symmetry group and exact solution to the (2+1)- dimemsional dissipative AKNS equation”, Romanian journal of physics, vol. 62, no. 5-6, Art. ID. 114, 2017. [On line]. Available: https://bit.ly/3b1wCTS
S. Sahoo and S. S. Ray, “Lie Symmetry analysis and exact solutions of (3+1) dimensional Yu-Toda-Sasa-Fukuyama equation in mathematical physics”, Computers & mathematics with applications, vol. 73, no. 2, pp. 253-260, Jan. 2017, doi: 10.1016/j.camwa.2016.11.016.
S. Yang and C. Hua, “Lie symmetry reductions and exact solutions of a coupled KdV-Burgers equation”, Applied mathematics and computation, vol. 234, pp. 579-583, May 2014, doi: 10.1016/j.amc.2014.01.044.
G. Wang, A. H. Kara, K. Fakhar, and J.V. Guzman, “Group analysis, exact solutions and conservation laws of a generalized fifth order KdV equation”, Chaos, solitons & fractals, vol. 86, pp. 8-15, May 2016, doi: 10.1016/j.chaos.2016.02.013.
R. Arora and A. Chauhan, “Lie symmetry reductions and solitary wave solutions of modified equal width wave equation”, International journal of applied and computational mathematics, vol. 4, no. 5, Art. ID. 122, Oct. 2018, doi: 10.1007/s40819-018-0557-z.
A. F. Cheviakov and J. F. Ganghoffer, “Symmetry properties of two dimensional Ciarlet-Mooney-Rivlin constitutive models in nonlinear elastodynamics”, Journal of mathematical analysis and applications, vol. 396, no. 2, pp. 625-639, Dec. 2012, doi: 10.1016/j.jmaa.2012.07.006.
Y. Yildirim, E. Yaşar, H. Triki, Q. Zhou, S.P. Moshokoa, M.Z. Ullah, A. Biswas, and M. Belić, “Lie symmetry analysis and exact solutions to N-coupled non-linear Schrödinger’s equations with kerr and parabolic law nonlinearities”, Romanian journal of physics, vol. 63, no. 1-2, Art. ID. 103, 2018. [On line]. Available: https://bit.ly/2RL01tO
H. Liu and J. Li, “Lie symmetry analysis and exact solutions for the short pulse equation”, Nonlinear analysis: theory, methods & applications, vol. 71, no. 5-6. pp. 2126-2133, Sep. 2009, doi: 10.1016/j.na.2009.01.075.
P. J. Olver, Applications of lie groups to differential equations. New York, NY: Springer-Verlag, 1993.
G. Bluman, A. Cheviakov, and S. Anco, Applications of symmetry methods to partial differential equations. New York, NY, 2010, doi: 10.1007/978-0-387-68028-6.
G. W. Bluman and J.D. Cole, Similarity methods for differential equations. New York, NY: Springer, 1974.
L. V. Ovsiannikov, Group analysis of differential equations. New York, NY: Academic Press, 1982.
N. Goyal, A. M. Wazwaz and R. K. Gupta, “Applications of maple software to derive exact solutions of generalized fifth-order Korteweg-De Vries equation with time-dependent coefficients”, Romanian reports in physics, vol. 68, no. 1, pp. 99-111, 2016. [On line]. Available: https://bit.ly/3aZaIR8
W. Malfliet, “Solitary wave solutions of nonlinear wave equations”, American journal of physics, vol. 60, no. 7, pp. 650-654, Jul. 1992, doi: 10.1119/1.17120.
A. M. Wazwaz, Partial differential equation and solitary wave theory, Berlin: Springer, 2009, doi: 10.1007/978-3-642-00251-9.
K. Raslan and Z. F. A. Shaeer, “The Tanh methods for the Hirota equations”, International journal of computer applications, vol. 107, no. 11, pp. 5–9, Dec. 2014, doi: 10.5120/18793-0134.
W. Rudin, Principles of mathematical analysis, 3rd ed., Beijing: China Machine Press, 2004.