On the new wave behavior of the Magneto-Electro-Elastic(MEE) circular rod longitudinal wave equation
The analytical solution of the longitudinal wave equation in the MEE circular rod is analyzed by the powerful sine-Gordon expansion method. Sine - Gordon expansion is based on the well-known wave transformation and sine - Gordon equation. In the longitudinal wave equation in mathematical physics, the transverse Poisson MEE circular rod is caused by the dispersion. Some solutions with complex, hyperbolic and trigonometric functions have been successfully implemented. Numerical simulations of all solutions are given by selecting the appropriate parameter values. The physical meaning of the analytical solution explaining some practical physical problems is given.
[1] Ren, Y.J., Liu S.T., and Zhang, H.Q.(2007). A new generalized algebra method and its application in the (2+1)-dimensional Boiti-Leon-Pempinelli equation. Chaos Soli-tons and Fractals, 32, 1655-1665.
[2] Manafian, J. (2016). Optical soliton solu- tions for Schr¨odinger type nonlinear evolu- tion equations by the tan(F(ξ)/2)-expansion method. Optik-International Journal of Light and Electron Optics, 127, 4222-4245.
[3] Feng, W.G., Li, K.M., Li, Y.Z. and Lin, C.(2009). Explicit exact solutions for (2+1)- dimensional Boiti-Leon-Pempinelli equation. Communications in Non-linear Science and Numerical Simulation, 14, 2013-2017.
[4] Fu, Z., Liu, S. and Zhao, Q. (2001) New Ja- cobi elliptic function expansion and new pe- riodic solutions of nonlinear wave equations. Physics Letters A, 290, 72-76.
[5] Zayed, E.M.E. and Alurrfi, K.A.E. (2014). The homogeneous balance method and its ap- plications for finding the exact solutions for nonlinear evolution equations. Italian Journal of Pure and Applied Mathematics, 33, 307- 318.
[6] Islam, M.S, Khan, K. and Arnous, A.H.(2015). Generalized Kudryashov method for solving some (3+1)-dimensional nonlinear evolution equations. New Trends in Mathe- matical Sciences, 3(3), 46-57.
[7] Lu, H.L., Liu, X.Q. and Niu, L. (2010). A generalized (G′ /G)-expansion method and its applications to nonlinear evolution equations. Applied Mathematics and Computation, 215, 3811-3816.
[8] Wang, C. (2016). Dynamic behavior of trav- eling waves for the Sharma-Tasso-Olver equa- tion. Non-linear Dynamics, 85(2), 1119-1126.
[9] Baskonus, H.M. and Bulut, H. (2016). Ex- ponential prototype structures for (2+1)- di- mensional Boiti-Leon-Pempinelli systems in mathematical physics. Waves in Random and Complex Media, 26(2), 201-208.
[10] Baskonus, H.M., Bulut, H. and Belgacem, F.B.M. (2017). Analytical Solutions for non- linear long-short wave interaction systems with highly complex structure. Journal of Computational and Applied Mathematics, 312, 257-266.
[11] Kadkhoda, N. and Jafari, H. (2016). Kudryashov method for exact solutions of isothermal magnetostatic atmospheres. Ap- plied Mathematics and Computation, 6(1), 43- 52.
[12] El-wakil, S.A., El-labany, S.K., Zahran, M.A. and Sabry, R. (2002). Modified extended tanh-function method for solving nonlinear partial diferential equations. Physics Letters A, 299, 179-188.
[13] Islam, M.S., Khan, K., Arnous, A.H. (2015). Generalized Kudryashov method for solving some (3+1)-dimensional nonlinear evolution equations. New Trends in Mathematical Sci- ences, 3(3), 46-57.
[14] Chen, H.T., Hong-Qing, Z. (2004). New dou- ble periodic and multiple soliton solutions of the generalized (2 + 1)-dimensional Boussi- nesq equation. Chaos Solitons and Fractals, 20, 765-769.
[15] Baskonus, H.M. and Bulut, H. (2015). On some new analytical solutions for The (2+1)- dimensional burgers equation and the special type of Dodd- Bullough-Mikhailov equation. Journal of Applied Analysis and Computa- tion, 5(4), 613-625.
[16] Lu, Z. and Zhang, H. (2004). Soliton like and multi-soliton like solutions for the Boiti- Leon-Pempinelli equation. Chaos, Solitons and Fractals, 19, 527-531.
[17] Huang, D.J. and Zhang, H.Q. (2004). Ex- act Travelling Wave Solutions for the Boiti- Leon-Pempinelli Equation. Chaos Solitons and Fractals, 22, 243-247.
[18] Dai, C. and Wang, Y. (2009). Periodic struc- tures based on variable separation solution of the (2+1)-dimensional Boiti-Leon-Pempinelli equation. Chaos Solitons and Fractals, 39, 350-355.
[19] Liang, Y. (2014). Exact solutions of the (3+1)-dimensional modified KdV- Zakharov- Kuznetsev equation and Fisher equations us- ing the modified simple equation method. Journal of Interdisciplinary Mathematics, 17, 565-578.
[20] Zhang, H. (2007). Extended jacobi elliptic function expansion method and its applica- tions. Communications in Non-linear Science and Numerical Simulation, 12(5), 627-635.
[21] Petrovic, N.Z. and Bohra, M. (2016). General jacobi elliptic function expansion method ap- plied to the generalized (3 + 1)-dimensional nonlinear Schrodinger equation. Optical and Quantum Electronics, 48, (268).
[22] Yan, Z. (2003). Jacobi elliptic function solu- tions of nonlinear wave equations via the new sinh-gordon equation expansion method. MM Research Preprints, 22, 363-375.
[23] Manafian, J. and Lakestani, M. (2019). Lump-type solutions and interaction phenom- enon to the bidirectional Sawada- Kotera equation. Pramana, 92:41.
[24] Manafian, J. (2018). Novel solitary wave so- lutions for the (3+1)-dimensional extended Jimbo-Miwa equations. Computers & Math- ematics with Applications, 76(5), 1246-1260.
[25] Manafian, J., Mohammadi-Ivatloo, B. and Abapour, M. (2019). Lump-type solutions and interaction phenomenon to the (2+1)- dimensional breaking soliton equation. Ap- plied Mathematics and Computation Volume, 356, 13-41.
[26] Ilhan, O.A., Manafian, J. and Shahriari M.(2019). Lump wave solutions and the inter- action phenomenon for a variable coe伍cient Kadomtsev-Petviashvili equation. Computers & Mathematics with Applications, In press.
[27] Foroutan, M., Manafian, J. and Ranjbaran, A. (2018). Lump solution and its interaction to (3+1)-D potential-YTSF equation. Nonlin- ear Dynamics, 92, 2077-2092.
[28] Mohammad, S. and Manafian, J. (2019). Analytical behaviour of lump solution and interaction phenomenon to the Kadomtsev- Petviashvili-like equation. Pramana, 93:3.
[29] Seadawy, A.R. and Manafian, J. (2018). New soliton solution to the longitudinal wave equa- tion in a magneto-electro-elastic circular rod. Results in Physics, 8, 1158-1167.
[30] Ramin, M.T., Manafian, J., Baskonus, H.M. and Du>sunceli, F. (2019). Applications of He’s semi-inverse variational method and ITEM to the nonlinear long-shortwave interaction sys- tem, IJAAS Journal, 6(8), 53-64.
[31] Yan, C. (1996). A simple transformation for nonlinear waves. Physics Letters A , 22(4), 77-84.
[32] Bulut, H., Sulaiman, T.A. and Baskonus, H.M. (2016). New solitary and optical wave structures to the Korteweg-de Vries equation with dual-power law nonlinearity. Optical and Quantum Electronics, 48:564, 1-14.
[33] Baskonus, H.M., Bulut, H. and Atangana, A. (2016). On the complex and hyperbolic structures of longitudinal wave equation in a Magneto-Electro-Elastic circular rod. Smart Materials and Structures, 25(3), 035022.
[34] Xue, C.X., Pan, E. and Zhang, X.Y. (2011). Solitary waves in a Magneto-Electro-Elastic circular rod. Smart Materials and Structures, 20(10), 035022.
[35] Ma, X., Pan, Y. and Chang, L. (2013). Ex- plicit travelling wave solutions in a Magneto- Electro-Elastic circular rod. International Journal of Computer Science Issues, 10(1), 62-68.
[36] Khan, K., Koppelaar, H. and Akbar, A.(2016). Exact and numerical soliton solutions to nonlinear wave equations. Computational and Mathematical Engineering, 2, 5-22.
[37] Younis, M. and Ali, S. (2015). Dark and sin- gular solitons in Magneto-Electro-Elastic cir- cular rod. Waves in Random and Complex Media, 25(4), 549-555.
[38] Yan, Z. and Zhang, H. (1999). New explicit and exact travelling wave solutions for a sys- tem of variant Boussinesq equations in math- ematical physics. Physics Letters A, 252, 291- 296
[39] Zhen-Ya, Y., Hong-oing, Z. and En-Gui, F.(1999). New explicit and travelling wave solu- tions for a class of nonlinear evolution equa- tions. Acta Physica Sinica, 48(1), 1-5.
[40] Weisstein, E.W. (2002). Concise encyclope- dia of nathematics. 2nd edition (New York: CRC Press).