The 2-D Boussinesq equation of 10th order is derived from its bilinear form. Its soliton solutions are studied in detail using the Hirota’s bilinear method. Since the 2-D Boussinesq equation is not completely integrable, we only obtain its 1-soliton and 2-soliton solutions. The equation is solved by the tanh method to reconstruct the 1-soliton solution obtained by the Hirota’s method.

higher order Boussinesq equation, Hirota bilinear method, tanh method.

Received: January 2, 2023; Revised: February 8, 2023; Accepted: February 18, 2023; Published: February 22, 2023

How to cite this article: K. Bharatha and R. Rangarajan, Soliton solutions of 10th order 2-D Boussinesq equation, Advances in Differential Equations and Control Processes 30(1) (2023), 73-82. http://dx.doi.org/10.17654/0974324323005

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, 1991.
[2] C. Baishya and R. Rangarajan, A new application of G'/G-expansion method for travelling wave solutions of fractional PDE’s, Int. J. Appl. Eng. Res. 13(11) (2018), 9936-9942.
[3] P. G. Drazin and R. S. Johnson, Solitons: An Introduction, Cambridge University Press, 1989.
[4] J. Hietarinta, A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV-type bilinear equations, J. Math. Phys. 28(8) (1987), 1732-1742.
[5] J. Hietarinta, Hirota’s bilinear method and its connection with integrability, Integrability, Springer, 2009, pp. 279-314.
[6] R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, 2004.
[7] R. S. Johnson, A two-dimensional Boussinesq equation for water waves and some of its solutions, J. Fluid. Mech. 323 (1996), 65-78.
[8] C. Liu and Z. Dai, Exact periodic solitary wave solutions for the - dimensional Boussinesq equation, J. Math. Anal. Appl. 367 (2010), 444-450.
[9] W. Malfliet, The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations, J. Comput. Appl. Math. 164 (2004), 529-541.
[10] W. Malfliet and W. Hereman, The tanh method. I. Exact solutions of nonlinear evolution and wave equations, Phys. Scripta 54(6) (1996), 563-568.
[11] W. Malfliet and W. Hereman, The tanh method. II. Perturbation technique for conservative systems, Phys. Scripta 54(6) (1996), 569-575.
[12] Y. Matsuno, Bilinear transformation method, Math. Sci. Eng. 174 (1984), 185-190.
[13] A. M. Wazwaz, Multiple-soliton solutions for the ninth-order KdV equation and sixth-order Boussinesq equation, Appl. Math. Comput. 206(2) (2008), 1005.
[14] A. M. Wazwaz, Partial differential equations and solitary waves theory, Nonlinear Physical Science, Higher Education Press, Beijing, Springer, Berlin, 2009.