Conservation laws and invariant solutions of time-dependent Calogero-Bogoyavlenskii-Schiff equation

Subject Areas : Differential geometry

Y. AryaNejad ¹ , R. Mirzavand ²

1 - Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Iran
2 - Institute of Advanced Studies, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran

Received: 2022-10-26 Accepted : 2022-12-26 Published : 2022-12-01

Keywords: Lie algebras, vCBS equation, reduction equations, conservation laws,

Abstract :

This paper uses the classical Lie method to determine symmetry reductions and exact solutions of the time-dependent Calogero-Bogoyavlenskii-Schiff equation (vCBS). This classical method generates some exact arbitrary solutions and exhibits various qualitative behaviors. Here, we derived the infinitesimal symmetries and six basic combinations of vector fields in the linear forms that can be utilized to transform the given equation into the PDEs with their variables. Further, we obtain comprehensive invariant solutions of the vCBS equation. Next, we apply a direct method to explore conservation laws. Finally, we determine the conservation laws of the vCBS equation via the Bluman-Anco homotopy formula.

References:

[1] S. C. Anco, M. L. Gandarias, E. Recio, Line-solitons, line-shocks, and conservation laws of a universal KP-like equation in 2+1 dimensions, J. Math. Anal. Appl. 504 (2021), 1:125319.

[2] Y. AryaNejad, Symmetry Analysis of wave equation on conformally Flat spaces, J. Geom. phys 161 (2021), 161:104029.

[3] Y. AryaNejad, Exact solutions of Diffusion Equation on sphere, Comput. Methods Differ. Equ. 10(3) (2022), 789-798.

[4] G.W. Bluman, S.C Anco, Symmetry and integration methods for differential equations, Springer, 2002.

[5] M. S. Bruzon, M. l. Gandarias, C. Muriel, J. Ramierez, G. S. Saez, F. R. Romero, The Calogero-Bogoyavlenskii-Schiff equation in (z+1) dimensions, Theor. Math. phys. 137 (1) (2003), 1367-1377.

[6] M. L.Gandarias, M. Torrisi, A. Valenti, Symmetry classification, and optimal systems of a non-linear wave equation, Internet. J. Nonlinear Mech. 39 (2004), 389-398.

[7] Y. N. Grigoriev, N. H. Ibrgimov, V. F. kovalev, S. V. Meleshko, Symmetries of Integro-Differential Equations: with Applications in Mechanics and Plasma Physics, Springer, 2010.

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

[9] AP. Màrquez, TM. Garrido, E. Recio, R. De la Rosa, Lie symmetries and exact solutions for a fourth-order nonlinear diffusion equation, Math. Meth. Appl. Sci. 45 (17) (2022), 10614-10627.

[10] A. Motamednezhad, F. Khajevand, Symmetry analysis and exact solutions of acoustic equation, Comput. Methods. Diff. Equ. 8 (3) (2020), 523-536.

[11] M. Nadjafikhah, Lie symmetries of inviscid burgers equations, Adv. Appl. Clifford Algebras. 19 (1) (2009), 701-712.

[12] L. V. Ovsiannikov, Group analysis of differential equations, Academic press, New York, 1982.

[13] P. J. Olver, P. Rosenau, Group-invariant solutions of differential equations, SIAM J. Appl. Math. 47 (2) (1987), 263-278.

[14] S. Rashidi, S. R. Hejazi, Self-adjointness, Conservation laws and invariant solutions of the Buckmaster equation, Comput. Methods Diff. Equ. 8 (1) (2020), 85-98.

[15] A. M. Wazwaz, A variety of completely integrable Calogero Bogoyavlenskii Schiff equation with time dependent coefficints, Inter. J. Numerical Methods. Heat. Fluid. Flow. 2020, online.