Document Type : Original Article

**Authors**

Department of Mathematics, Payame Noor University, PO BOX 19395-4697, Tehran, Iran

**Abstract**

In this paper, we prove that the perturbed generalized BenjaminBona-Mahony (BBM) equation with a small parameter is approximately nonlinear self-adjoint. It’s important for constructing approximate conservation laws associated with approximate symmetries.

**Keywords**

- Perturbed generalized generalized Benjamin-Bona-Mahony (BBM) equation
- Approximate symmetry
- Approximate conservation laws
- Self-adjoint

**Main Subjects**

[1] R. L. Anderson, V. A. Baikov, R. K. Gazizov, W. Hereman, N. H. Ibragimov, F. M. Mahomed, S. V. Meleshko, M. C. Nucci, P. J. Olver, M. B. Sheftel’, A. V. Turbiner, and E. M. Vorob’ev, CRC handbook of Lie group analysis of differential equations. Vol. 3, CRC Press, Boca Raton, FL, 1996. New trends in theoretical developments and computational methods.

[2] V. A. Ba˘ıkov, R. K. Gazizov, and N. K. Ibragimov, Approximate symmetries, Mat. Sb. (N.S.), 136(178) (1988), pp. 435–450, 590.

[3] , Perturbation methods in group analysis, in Current problems in mathematics. Newest results, Vol. 34 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989, pp. 85–147, 195. Translated in J. Soviet Math. 55 (1991), no. 1, 1450–1490.

[4] T. B. Benjamin, Lectures on nonlinear wave motion, in Nonlinear wave motion (Proc. AMS-SIAM Summer Sem., Clarkson Coll. Tech., Potsdam, N.Y., 1972), Lectures in Appl. Math., Vol. 15, Amer. Math. Soc., Providence, RI, 1974, pp. 3–47.

[5] T. B. Benjamin, J. L. Bona, and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), pp. 47–78.

[6] J. Biazar and Z. Ayati, Extension of the Exp-function method for systems of two-dimensional Burgers equations, Comput. Math. Appl., 58 (2009), pp. 2103–2106.

[7] G. Derks and S. van Gils, On the uniqueness of traveling waves in perturbed Korteweg-de Vries equations, Japan J. Indust. Appl. Math., 10 (1993), pp. 413–430.

[8] B. A. M. M. A. Y. Fadhil H. Easif, Saad A. Manaa, Variational homotopy perturbation method for solving Benjamin-Bona-Mahony equation, Appl. Math., 6 (2015), pp. 675–683.

[9] W. I. Fushchich and W. M. Shtelen, On approximate symmetry and approximate solutions of the nonlinear wave equation with a small parameter, J. Phys. A, 22 (1989), pp. L887–L890.

[10] S. R. Hejazi and A. Naderifard, Dym equation: group analysis and conservation laws, AUT J. Math. Com., 3 (2022), pp. 17–26.

[11] N. H. Ibragimov and V. F. Kovalev, Approximate and renormgroup symmetries, Nonlinear Physical Science, Springer, Berlin; Higher Education Press, Beijing, 2009.

[12] M. Jafari and R. Darvazebanzade, Approximate symmetry group analysis and similarity reductions of the perturbed mKdV-KS equation, Comput. Methods Differ. Equ., 11 (2023), pp. 175–182.

[13] M. Jafari and S. Mahdion, Non-classical symmetry and new exact solutions of the Kudryashov-Sinelshchikov and modified KdV-ZK equations, AUT J. Math. Com., 4 (2023), pp. 195–203.

[14] M. Jafari, A. Zaeim, and M. Gandom, On similarity reductions and conservation laws of the two nonlinearity terms Benjamin-Bona-Mahony equation, J. Math. Ext., 17 (2023), pp. 1–22.

[15] M. Jafari, A. Zaeim, and A. Tanhaeivash, Symmetry group analysis and conservation laws of the potential modified KdV equation using the scaling method, Int. J. Geom. Methods Mod. Phys., 19 (2022), pp. Paper No. 2250098, 14.

[16] S. Lie, Integration of a class of linear partial differential equations by means of definite integrals, in Lie group analysis: Classical heritage. Translated by Nail H. Ibragimov, Elena D. Ishmakova, Roza M. Yakushina, Karlskrona: ALGA, Blekinge Institute of Technology, 2004, pp. 65–102.

[17] S. Mbusi, B. Muatjetjeja, and A. R. Adem, On the exact solutions and conservation laws of a generalized (1 + 2)-dimensional Jaulent-Miodek equation with a power law nonlinearity, Int. J. Nonlinear Anal. Appl., 13 (2022), pp. 1721–1735.

[18] M. Nadjafikhah and O. Chekini, Conservation law and Lie symmetry analysis of foam drainage equation, AUT J. Math. Com., 2 (2021), pp. 37–44.

[19] M. Nadjafikhah and M. Jafari, Computation of partially invariant solutions for the Einstein Walker manifolds’ identifying equations, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), pp. 3317–3324.

[20] E. Noether, Invariant variational problems, Nachr. Ges. Wiss. G¨ottingen, Math.-Phys. Kl., 1918 (1918), pp. 235–257.

[21] T. Ogawa, Travelling wave solutions to a perturbed Korteweg-de Vries equation, Hiroshima Math. J., 24 (1994), pp. 401–422.

[22] P. J. Olver, Applications of Lie groups to differential equations, vol. 107 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1986.

[23] M. Wang, Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A, 213 (1996), pp. 279–287.

[24] A.-M. Wazwaz, Exact solutions with compact and noncompact structures for the one-dimensional generalized benjamin–bona–mahony equation, Commun. Nonlinear Sci. Numer. Simul., 10 (2005), pp. 855–867.

[25] W. Yan, Z. Liu, and Y. Liang, Existence of solitary waves and periodic waves to a perturbed generalized KdV equation, Math. Model. Anal., 19 (2014), pp. 537–555.

[26] S. Yang and C. Hua, Lie symmetry reductions and exact solutions of a coupled KdV-Burgers equation, Appl. Math. Comput., 234 (2014), pp. 579–583.

[27] M. Younis, A new approach for the exact solutions of nonlinear equations of fractional order via modified simple equation method, Appl. Math., 5 (2014), pp. 1927–1932.

[28] Zhaqilao and Z. Qiao, Darboux transformation and explicit solutions for two integrable equations, J. Math. Anal. Appl., 380 (2011), pp. 794–806.

January 2024

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