Densities and fluxes of the conservation laws for the Kuramoto-Sivashinsky equation

محورهای موضوعی : Geometry

M. Jafari ¹ , Y. Alipour Fakhri ² , M. Khadivar ³

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

تاریخ دریافت : 1400/11/15 تاریخ پذیرش : 1401/01/04 تاریخ انتشار : 1401/01/05

کلید واژه: KS equation‎, ‎Conversation laws‎, ‎scaling symmetry‎, ‎homotopy operator,

چکیده مقاله :

‎In this paper‎, ‎the main purpose is to calculate the conservation laws of Kuramoto-Sivashinsky equation using‎ the scaling method‎. ‎Linear algebra and calculus of variations are used in this algorithmic method‎. ‎Also the density of the conservation law is obtained by scaling symmetries of the equation and the flux corresponding to the density is calculated using the homotopy operator‎.

منابع و مأخذ:

[1] Z. Ali, S. Husnine, I. Naeem, Exact solutions of generalized modified Boussinesq, Kuramoto-Sivashinsky and Camassa-Holm Equations via Double Reduction Theory, J. Appl. Math. (2013), 2013:902128.

[2] S. Anco, G. Bluman, Direct construction method for conservation laws of partial differential equations Part II: general treatment, Euro. J. Appl. Math. 13 (2002), 567-585.

[3] F. Calogero, A. Degasperis, Spectral Transform and Solitons, North Holland, Amsterdam, 1982.

[4] A. F. Cheviakov, Computation of fluxes of conservation laws, J. Engin. Math. 66 (2010), 153-173.

[5] W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman, B. M. Herbst, Direct Methods and Symbolic Software for Conservation Laws of Nonlinear equations, Advances in Nonlinear Waves and Symbolic Computation, Nova Science Publishers, New York, 2009.

[6] M. Jafari, A. Tanhaeivash, Symmetry group analysis and similarity reductions of the thin film equation, J. Linear. Topol. Algebra. 10 (4) (2021), 287-293.

[7] M. Jafari, A. Zaeim, S. Mahdion, Scaling symmetry and a new conservation law of the Harry Dym equation, Mathematics Interdisciplinary Research. 6 (2) (2021), 151-158.

[8] A. Kalogirou, E. E. Keaveny, D. T. Papageorgiou, An in-depth numerical study of the two-dimensional Kuramoto-Sivashinsky equation, Proc. Royal. Soc. A. Math. Phy. Engin. Sci. (2015), 471:20140932.

[9] M. Nadjafikhah, R. Bakhshandeh Chamazkoti, Cartan equivalence problem for third-order differential operators, Turkish J. Math. 37 (6) (2013), 949-958.

[10] M. Nadjafikhah, M. Jafari, Computation of partially invariant solutions for the Einstein Walker manifolds' identifying equations, Commun. Nonlinear Sci. Numer. Simul. 18 (12) (2013), 3317-3324.

[11] E. Noether, Invariante variations problem, Nachr. Akad. Wiss. G¨ott. Math. Phys. Kl. 2 (1918), 235-257.

[12] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1986.

[13] P. Paolo, M. Chaouqi, Modified Kuramoto-Sivashinsky equation: Stability of stationary solutions and the consequent dynamics, Phys. Rev. E. (2007), 75:027202.

[14] D. Poole, Symbolic Computation of Conservation Laws of Nonlinear Partial Differential Equations using Homotopy Operators, Ph.D. dissertation, Colorado School of Mines, Golden, Colorado, 2009.

[15] D. Poole, W. Hereman, The homotopy operator method for symbolic integration by parts and inversion of divergences with applications, Appl. Anal. 87 (2010), 433-455.

[16] M. Rosa, J. C. Camacho, M. Bruz´on, M. L. Gandarias, Lie symmetries and conservation laws for a generalized Kuramoto-Sivashinsky equation, Math. Meth. Appl. Sci. 41 (17) (2018), 7295-7303.

[17] V. Volterra, Lecons sur les Fonctions de Lignes, Gauthier-Villars, Paris, 1913.

[18] A. Wazwaz, New solitary wave solutions to the Kuramoto-Sivashinsky and the Kawahara equations, Appl. Math. Comput. 182 (2006), 1642-1650.