Document Type : Original Article

**Author**

Department of Mathematics, Broujerd Branch, Islamic Azad University, Broujerd, Iran

**Abstract**

The goal of this article is to compute conservation law, Lagrangian and $\mu$-conservation law of the generalized Rosenau-type equation using the homotopy operator, the $\mu$-symmetry method and the variational problem method. The generalized Rosenau-type equation includes the generalized Rosenau equation, the generalized Rosenau-RLW equation and the generalized Rosenau-KdV equation, which admits the third-order Lagrangian. The article also compares the conservation law and the $\mu$-conservation law of these three equation.

**Keywords**

[2] A. Esfahani, Solitary wave solutions for generalized Rosenau-KdV equation, Commun. Theor. Phys., 55

(2011), p. 396.

[3] G. Gaeta and P. Morando, On the geometry of lambda-symmetries and PDE reduction, J. Phys. A: Math.

Gen., 37 (2004), p. 6955.

[4] D. J. Korteweg and 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), pp. 422–443.

[5] C. Muriel, J. L. Romero, and P. J. Olver, Variational C∞-symmetries and Euler-Lagrange equations,

J. Differential Equations, 222 (2006), pp. 164–184.

[6] P. J. Olver, Applications of Lie groups to differential equations, vol. 107 of Graduate Texts in Mathematics,

Springer-Verlag, New York, 1986.

[7] X. Pan and L. Zhang, On the convergence of a conservative numerical scheme for the usual Rosenau-RLW

equation, Appl. Math. Model., 36 (2012), pp. 3371–3378.

[8] D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 25 (1966), pp. 321–330.

[9] P. Rosenau, A quasi-continuous description of a nonlinear transmission line, Phys. Scr., 34 (1986), p. 827.

[10] P. Rosenau, Dynamics of dense discrete systems: High order effects, Prog. Theor. Phys., 79 (1988), pp. 1028–

1042.

[11] J.-M. Zuo, Y.-M. Zhang, T.-D. Zhang, and F. Chang, A new conservative difference scheme for the

general Rosenau-RLW equation, Bound. Value Probl., (2010), pp. Art. ID 516260, 13.

July 2024

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