Order Reduction, $\mu$-Symmetry and $\mu$-Conservation Law of The Generalized mKdV Equation with Constant-coefficients and Variable-‎coefficients‎

Subject Areas : International Journal of Industrial Mathematics

خدایار گودرزی ¹

1 - Department of Mathematics, Borujerd Branch, Islamic Azad University, Borujerd, Iran.

Received: 2021-10-29 Accepted : 2022-06-25 Published : 2022-09-01

Keywords: Order ‎reduction‎, $mu$-symmetry, S‎ymmetry, Variational problem, $mu$-conservation law,

Abstract :

The goal of this paper is to calculate the order reduction of the generalized mKdV equation with constant coefficients (gmKdVcc) and the generalized mKdV equation with variable-coefficients (gmKdVvc) using the mu-symmetry method. Moreover we obtain Lagrangian and mu-conservation law of the gmKdVcc equation and the gmKdVvc equation using the variational problem method.

References:

 [1] H. Airault, Rational solutions of Painlev´e equation, Studies in Applied Mathematics 61 (1979) 31-53.
[2] A. Biswas, Solitary wave solution for the generalized KdV equation with time- dependent damping and dispersion,  Commun, Nonlinear Sci. Numer. Simulat 14 (2009) 3503-3506.
[3] T. B. Benjamin, J. L. Bona, J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Trans. R. Soc. (Lond) ser.A 272 (1992).
[4] W. Bluman, F. Cheviakov, C. Anco, Construction of conservation law: how the direct method generalizes Nother’s theorem, Group analysis of differential equations and integrability (2009) 1-23.
[5] J. L. Bona, P. J. Bryant, A mathematical model for long waves gener- ated by wave makers in nonlinear dispersive systems, Proc. Cambridge Phil. Soc (1973).
[6] H. H. Chen, C. S. Liu, Solitons in nonuniform media, Phys. Rev. Lett 37 (1976).
[7] W. L. Chan, L. Kam-Shun, Nonpropagating solitons of the variable coefficient and nonisospectral Kortewegde Vries equation, J. Math. Phys 30 (1989) 2521-2526.
[8] G. Cicogna, G. Gaeta, P. Morando, On the relation between standard and µ-symmetries for PDEs, J. Phys. A 37 (2004) 9467-9486.
[9] G. Cicogna, G. Gaeta, Norther theorem for µ-symmetries, J. Phys.A 40 (2007) 11899-11921.
[10] P. G. Drazin, R. S. Johnson, Solitons: An Introduction, Cambridge University press, London (1983).
[11] Z. T. Fu, S. D. Liu, S. K. Liu, New exact solutions to KdV equations with variable coefficients or forcing, Appl. Math. Mech 25 (2004) 73-79.
[12] G. Gaeta, P. Morando, On the geometry of lambda-symmetries and PDEs reduction, J. Phys. A 37 (2004) 6955-6975.
[13] G. Gaeta, Lambda and mu-symmetries,

[14] KH. Goodarzi, M. Nadjafikhah, µ-symmetry and µ-conservation law for the extended mKdV equation, JNMP 3 (2014) 371-381.
[15] D. J.Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Phil. Mag 39 (1895) 422-443.
[16] A. Kudryashov, I. Sinelshchikov, A note on the Lie symmetry analysis and exact solutions for the extended mKdV equation, Acta. Appl.Math 113 (2011) 41-44.
[17] E. V. Krishnan, Y. Peng, Exact solutions to the combined KdV-mKdV equation by the extended mapping method, Phys. Scr 73 (2006) 405-409.
[18] C. Muriel, J. L. Romero, New methods of reduction for ordinary differential equation, IMA J. Appl. Math 66 (2001) 111-125.
[19] C. Muriel, J. L. Romero, C1-symmetries and reduction of equation without Lie point symmetries, J. Lie Theory 13 (2003) 167-188.
[20] C. Muriel, J. L. Romero, P.J. Olver, Variationl C1-symmetries and Euler-Lagrange equations, J. Diff. Eqs 222 (2006) 164-184.
[21] C. Muriel, J. L. Romero, Prolongations of vector fields and the invariants-byderrivation property, Theor. Math. Phys 133 (2002) 1565-1575.
[22] P. J. Olver, Applications of Lie Groups to Differential Equations, New York (1986).
[23] C. Vasconcellos, P. N. da Silva, Stabilization of the linear Kawahara equation, Applied and Computational Mathematics 3 (2015).
[24] ML.Wang, YM. Wang, Phys. Lett (2001) 287- 291.
[25] T. Xiao-Yan, H. Fei, L. Sen-Yue, Variable coecient KdV equation and the analytical diagnosis of a dipole blocking life cycle, Chinese Phys. Lett 23 (2006) 887-890.
[26] X. Yu, Y. T. Gao, Z. Y. Sun, Y. Liu. Solitonic propagation and interaction for a generalized variable-coefficient forced Kortewegde Vries equation in fluids, J. Phys. Rev. E 83 (2011).
[27] X. Yu, Y. T. Gao, Z. Y. Sun, Y. Liu. N-soliton solutions, Bcklund transformation and Lax pair for a generalized variable oefficient fifth-order Korteweg-de Vries equation, J.Phys. Scr 81 (2010).
[28] S. Zhang, Application of Exp-function method to a KdV equation with variable coefficients, Phys. Lett.A 365 (2007) 448-453.