Abstract

We study a kdv-burgers equation by Lie group analysis. We obtain Lie point symmetries and use them to carry out symmetry reductions. The arising systems are investigated for exact solutions. Solitons are constructed by a linear span of time and space translation symmetries. We also compute conservation laws using multiplier approach.

Keywords

References

[1] Bhrawy, A. H., Zaky, M. A., and Baleanu, D. New numerical approximations for space-time fractional burgers' equations via a legendre spectral-collocation method. Rom. Rep. Phys, 67 (2):340-349, 2015.
[2] Bluman, G. W. and Kumei, S. Symmetries and differential equations, volume 81. Springer Science & Business Media, 1989.
[3] Bluman, G. W., Cheviakov, A. F., and Anco, S. C. Applications of symmetry methods to partial differential equations, volume 168. Springer, 2010.
[4] Demiray, H. and Antar, N. Nonlinear waves in an inviscid fluid contained in a prestressed viscoelastic thin tube. Zeitschrift für angewandte Mathematik und Physik ZAMP, 48(2):325- 340, 1997.
[5] Gardner, C. S., Greene, J. M., Kruskal, M. D., and Miura, R. M. Korteweg-devries equation and generalizations. vi. methods for exact solution. Communications on pure and applied mathematics, 27(1):97-133, 1974.
[6] Golmankhaneh, A. K., Golmankhaneh, A. K., and Baleanu, D. Homotopy perturbation method for solving a system of schrödinger-korteweg-de vries equations. Rom. Rep. Phys, 63(3):609-623, 2011.
[7] Hassan, M. Exact solitary wave solutions for a generalized kdv-burgers equation. Chaos, Solitons & Fractals, 19(5):1201-1206, 2004.
[8] Ibragimov, N. H. Selected works. Volume 1-4. ALGA publications, Blekinge Institute of Technology, 2006-2009.
[9] Ibragimov, N. H. A new conservation theorem. Journal of Mathematical Analysis and Applications, 333(1):311-328, 2007.
[10] Ibragimov, N. H. A Practical Course in Differential Equations and Mathematical Modelling: Classical and New Methods. Nonlinear Mathematical Models. Symmetry and Invariance Principles. World Scientific Publishing Company, 2009.
[11] Johnson, R. A non-linear equation incorporating damping and dispersion. Journal of Fluid Mechanics, 42(1):49-60, 1970.
[12] Johnson, R. Shallow water waves on a viscous fluid the undular bore. The Physics of Fluids, 15(10):1693-1699, 1972.
[13] Lie, S. Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen. BG Teubner, 1891.
[14] Noether, E. Invariant variations problem. Nachr. Konig. Gissel. Wissen, Gottingen. Math. Phys. Kl, pages 235-257, 1918.
[15] Olver, P. J. Applications of Lie groups to differential equations, volume 107. Springer Science & Business Media, 1993.
[16] Ovsyannikov, L. Lectures on the theory of group properties of differential equations. World Scientific Publishing Company, 2013.
[17] Owino, J. O. A group approach to exact solutions and conservation laws of classical burger's equation. International Journal of Mathematics And Computer Research, 10(9):2894-2909, 2022.
[18] Owino, J. O. Group invariant solutions and conserved vectors for a special kdv type equation. International Journal of Advanced Multidisciplinary Research and Studies, 2(5):9-26, 2022.
[19] Owino, J. O. An application of lie point symmetries in the study of potential burger's equation. International Journal of Advanced Multidisciplinary Research and Studies, 2(5):191-207, 2022.
[20] Owino, J. O. and Okelo, B. Lie group analysis of a nonlinear coupled system of korteweg-de vries equations. European Journal of Mathematical Analysis, 1:133-150, 2021.
[21] Owuor, J. Conserved quantities of a nonlinear coupled system of korteweg-de vries equations. International Journal of Mathematics And Computer Research, 10(5):2673-2681, 2022.
[22] Owuor, J. Exact symmetry reduction solutions of a nonlinear coupled system of korteweg-de vries equations. International Journal of Advanced Multidisciplinary Research and Studies, 2 (3):76-87, 2022.
[23] Owuor, J. Group analysis on one-dimensional heat equation. International Journal of Advanced Multidisciplinary Research and Studies, 2(5):525-540, 2022.
[24] P.E, H. Symmetry methods for differential equations: a beginner's guide, volume 22. Cambridge University Press, 2000.
[25] Ruderman, M. Method of derivation of the korteweg-de vries-burgers equation: Pmm vol. 39, n= 4, 1975, pp. 686-694. Journal of Applied Mathematics and Mechanics, 39(4):656-664, 1975.
[26] Shi, Y., Xu, B., and Guo, Y. Numerical solution of korteweg-de vries-burgers equation by the compact-type cip method. Advances in Dierence Equations, 2015(1):1-9, 2015.
[27] Soliman, A. A numerical simulation and explicit solutions of kdv-burgers' and lax's seventhorder kdv equations. Chaos, Solitons & Fractals, 29(2):294-302, 2006.
[28] Soliman, A. Exact solutions of kdv-burgers' equation by exp-function method. Chaos, Solitons & Fractals, 41(2):1034-1039, 2009.
[29] Su, C. H. and Gardner, C. S. Korteweg-de vries equation and generalizations. iii. derivation of the korteweg-de vries equation and burgers equation. Journal of Mathematical Physics, 10 (3):536-539, 1969.
[30] Van Wijngaarden, L. On the motion of gas bubbles in a perfect fluid. Ann. Rev. Fluid Mech, 4:369-373, 1972.
[31] Wang, M. Exact solutions for a compound kdv-burgers equation. Physics Letters A, 213(5-6): 279-287, 1996.
[32] Wang, Q. Homotopy perturbation method for fractional kdv-burgers equation. Chaos, Solitons & Fractals, 35(5):843-850, 2008.
[33] Wazwaz, A.-M. Partial differential equations and solitary waves theory. Springer Science & Business Media, 2010.