eesti teaduste
akadeemia kirjastus
SINCE 1952
Proceeding cover
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
Impact Factor (2022): 0.9
Solitary waves, shock waves and conservation laws with the surface tension effect in the Boussinesq equation; pp. 17–29

Anjan Biswas, Jose Vega-Guzman, Anupma Bansal, Abdul H. Kara, Maggie Aphane, Yakup Yıldırım, Hashim M. Alshehri

This paper secures solitary waves, shock waves and singular solitary waves for the Boussinesq equation, which is studied with the inclusion of surface tension. The method of undetermined coefficients has yielded such waves. The Lie symmetry analysis has introduced a fresh perspective to the model. Conserved densities and corresponding conserved quantities are computed using the multiplier approach.


1. Anco, S. and Bluman, G. Direct construction method for conservation laws of partial differential equations Part I: Examples of conservation law classifications. Eur. J. Appl. Math., 2002, 13(5), 545–566.

2. Anderson, I. M. and Pohjanpelt, J. The cohomology of invariant variational bicomplexes. Acta Appl. Math., 1995, 41, 3–19.

3. Daripa, P. Higher-order Boussinesq equations for two-way propagation of shallow water waves. Eur. J. Mech. B Fluids, 2006, 25(6), 1008–1021.

4. Guo, C. Global existence and pointwise estimates of solutions for the generalized sixth-order Boussinesq equation. Commun. Math. Sci., 2017, 15(5), 1457–1487.

5. Jaharuddin. Traveling wave solutions for the nonlinear Boussinesq water wave equation. Glob. J. Pure Appl. Math., 2016, 12(1), 87–93.

6. Kolkovska, N. and Vucheva, V. Numerical investigation of sixth-order Boussinesq equation. AIP Conf. Proc., 2017, 1895, 110003.

7. Polat, N. and Piskin, E. Existence and asymptotic behavior of solution of Cauchy problem for the damped sixth-order Boussinesq equation. Acta. Math. Appl. Sin., 2015, 31(3), 735–746.

8. Recio, E., Gandarias, M. L. and Bruzon, M. S. Symmetries and conservation laws for a sixth-order Boussinesq equation. Chaos Solit. Fractals, 2016, 89, 572–577.

9. Taskesen, H. and Polat, N. Global existence for a double dispersive sixth-order Boussinesq equation. Contemp. Anal. Appl. Math., 2013, 1(1), 60–69.

10. Yokus, A. and Kaya, D. Conservation laws and a new expansion method for sixth-order Boussinesq equation. AIP Conf. Proc., 2015, 1676, 020062.

11. Olver, P. J. Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics. Springer, New York, NY, 1993.

12. Bansal, A., Biswas, A., Triki, H., Zhou, Q., Moshokoa, S. P. and Belic, M. Optical solitons and group invariant solutions to Lakhshmanan–Porsezian–Daniel model in optical fibers and PCF. Optik, 2018, 160, 86–91.

13. Bansal, A., Biswas, A., Mahmood, M. F., Zhou, Q., Mirzazadeh, M., Alshomrani, A. S., Moshokoa, S. P. and Belic, M. Optical soliton perturbation with Radhakrishnan–Kundu–Lakshmanan equation by Lie group analysis. Optik, 2018, 163, 137–141.

14. Kumar, S. Invariant solutions and bifurcation analysis of the nonlinear transmission line model. Nonlinear Dyn., 2021, 106(1), 211–227.

15. Wazwaz, A.-M. Painlevé analysis for Boiti–Leon–Manna–Pempinelli equation of higher dimensions with time-dependent coefficients: Multiple soliton solutions. Phys. Lett. A, 2020, 384(16), 126310.

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