New Traveling Wave Solutions for the Sixth-order Boussinesq Equation

In this paper, we investigate the new traveling wave solutions for the sixth-order Boussinesq equation using the tanh-coth method. Such a method is a type of expansion method that represents the solutions of partial differential equations as polynomials of $\tanh$ and $\coth$ functions. We discover several new traveling wave solutions for the sixth-order Boussinesq equation with different parameters, which are of fundamental importance for various applications.

Keywords:

Boussinesq equation, tanh-coth method Traveling wave solution,

PDF

___

[1] J. Boussinesq, Theorie de I’intumescence liquide, applelee onde solitaire ou de translation, se propageant dans un canal rectangulaire, C. R. Acad. Sci., 72 (1871), 755-759.
[2] J. Boussinesq, Theorie des ondes et des remous qui se progagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl., 17 (1872), 55-108.
[3] J. Zhou, H. Zhang, Well-posedness of solutions for the sixth-order Boussinesq equation with linear strong damping and nonlinear source, J. Nonlinear Sci., 31 (2021), 1-61.
[4] J. Liu, X. Wang, J. Zhou, H. Zhang, Blow-up phenomena for the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source, Discrete Contin. Dyn. Syst. - S, 14 (2021), 4321-4335.
[5] H. Wang, A. Esfahani, Global rough solutions to the sixth-order Boussinesq equation, Nonlinear Anal. Theory Methods Appl., 102 (2014), 97-104.
[6] H. Yang, An inverse problem for the sixth-order linear Boussinesq-type equation, UPB Sci. Bull. A: Appl. Math. Phys., 82 (2020), 27-36.
[7] C.I. Christov, G.A. Maugin, M.G. Velarde, Well-posed Boussinesq paradigm with purely spatial higher-order derivatives, Phys. Rev. E, 54 (1996), 3621-3638.
[8] S. Li, M. Chen, B.Y. Zhang, Exact controllability and stability of the sixth order Boussinesq equation, (2018), arXiv:1811.05943 [math.AP].
[9] S. Amirov, M. Anutgan, Analytical solitary wave solutions for the nonlinear analogues of the Boussinesq and sixth-order modified Boussinesq equations, J. Appl. Anal. Comput., 7 (2017), 1613-1623.
[10] A.M. Wazwaz, The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations, Appl. Math. Comput., 188 (2007), 1467-1475.
[11] O.F. Gozukizil, S. Akcagil, The tanh-coth method for some nonlinear pseudoparabolic equations with exact solutions, Adv. Differ. Equ., 2013 (2013), Article ID 143, 18 pages, doi: 10.1186/1687-1847-2013-143.
[12] F. Al-Askar, W. Mohammed, A. Albalahi, M. El-Morshedy, The impact of the Wiener process on the analytical solutions of the stochastic (2+1)-dimensional breaking soliton equation by using tanh-coth method, Math., 10 (2022), Article ID 817, 9 pages, doi: 10.3390/math10050817.
[13] W. Mohammed, M. El-Morshedy, The influence of multiplicative noise on the stochastic exact solutions of the Nizhnik-Novikov-Veselov system, Math. Comput. Simul., 190 (2021), 192-202.
[14] A. Rani, A. Zulfiqar, J. Ahmad, Q. Hassan, New soliton wave structures of fractional Gilson-Pickering equation using tanh-coth method and their applications, Results Phys., 29 (2021), Article ID 104724, 14 pages, doi: 10.1016/j.rinp.2021.104724.
[15] A. Mamum, S. Ananna, T. An, M. Asaduzzaman, M. Miah, Solitary wave structures of a family of 3D fractional WBBM equation via the tanh-coth approach, Partial Differ. Equ. Appl. Math., 5 (2022), Article ID 100237, 6 pages, doi: 10.1016/j.padiff.2021.100237.
[16] A.M. Wazwaz, New traveling wave solutions to the Boussinesq and the Klein-Gordon equations, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 889-901.
[17] M.T. Darvish, S. Arbabi, M. Najafi, A.M. Wazwaz, Traveling wave solutions of a (2+1)-dimensional Zakharov-like equation by the first integral method and the tanh method, Optik, 127 (2016), 6312-6321.
[18] S. Akcagil, T. Aydemir, New exact solutions for the Khokhlov-Zabolotskaya-Kuznetsov, the Newell-Whitehead-Segel and the Rabinovich wave equations by using a new modification of the tanh coth method, Cogent Math., 3 (2016), Article ID 1193104, 12 pages, doi: 10.1080/23311835.2016.1193104.
[19] Z. Yang, W. Zhong, W. Zhong, M. Belic, New traveling wave and soliton solutions of the sine-Gordon equation with a variable coefficient, Optik, 198 (2019), Article ID 163247, 5 pages, doi: 10.1016/j.ijleo.2019.163247.
[20] J. Fang, C. Dai, Optical solitons of a time-fractional higher-order nonlinear Schrodinger equation, Optik, 209 (2020), Article ID 1645574, doi: 10.1016/j.ijleo.2020.164574.
[21] A.R. Seadawy, K. El-Rashidy, Traveling wave solutions for some coupled nonlinear evolution equations, Math. Comput. Model., 57 (2013), 1371-1379.
[22] M. Najafi, S. Arbabi, Traveling wave solutions for nonlinear Schrodinger equations, Optik, 126 (2015), 3992-3997.
[23] A. Bansal, R. Gupta, Modified (G0=G)-expansion method for finding exact wave solutions of the coupled Klein-Gordon-Schrodinger equation, Math. Methods Appl. Sci., 35 (2012), 1175-1187.
[24] A.M. Wazwaz, The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations, Appl. Math. Comput., 167 (2005), 1196-1210.