Scaling Symmetry and a New Conservation Law of the Harry Dym Equation

Document Type : Original Scientific Paper


Department of Mathematics, Payame Noor University, P. O. BOX 19395-3697, Tehran, Iran



In this paper, we obtain a new conservation law for the Harry Dym equation by using the scaling method. This method is algorithmic and based on variational calculus and linear algebra. In this method, the density of the conservation law is constructed by considering the scaling symmetry of the equation and the associated flux is obtained by the homotopy operator. This density-flux pair gives a conservation law for the equation. A conservation law of rank 7 is constructed for the Harry Dym equation.


[1] S. Anco and G. Bluman, Direct construction method for conservation laws of partial differential equations Part II: general treatment, Euro. J. Appl. Math. 13 (5) (2002) 567–585.
[2] M. Antonowicz and A. P. Fordy, Coupled Harry Dym equations with multi-Hamiltonian structures, J. Phys. A: Math. Gen. 21 (5) (1988) L269–L275.
[3] G. W. Bluman, S. C. Anco, Symmetry and Integration Methods for Differential Equations, Springer-Verlag, New York, 2004.
[4] G. W. Bluman, A. F. Cheviakov and S. C. Anco, Applications of Symmetry Methods to Partial Differential Equations, Appl. Math. Sci., vol. 168, Springer-Verlag, New York, 2010.
[5] A. F. Cheviakov, Computation of fluxes of conservation laws, J. Eng. Math. 66 (1-3) (2010) 153–173.
[6] A. F. Cheviakov, GeM software package for computation of symmetries and conservation laws of differential equations, Comp. Phys. Comm. 176 (1) (2007) 18–61.
[7] W. Hereman, P. J. Adams, H. L. Eklund, M. S Hickman and B. M. Herbst, Direct methods and symbolic software for conservation laws of nonlinear equations, Advances in nonlinear waves and symbolic computation, 19 − 78, loose
errata, Nova Sci. Publ., New York, 2009.
[8] W. Hereman, P. P. Banerjee and M. R. Chatterjee, Derivation and implicit solution of the Harry Dym equation and its connections with the Korteweg–de Vries equation, J. Phys. A 22 (1989) 241–255.
[9] M. D. Kruskal, Nonlinear wave equations, Lecture Notes in Physics, vol. 38, pp. 310–354, Berlin-Heidelberg-New York: Springer, 1975.
[10] F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (5) (1978) 1156–1162.
[11] M. Nadjafikhah and P. Kabi-Nejad, On the changes of variables associated with the Hamiltonian structure of the Harry-Dym equation, Glob. J. Adv. Res. Cl. Mod. Geom. 6 (2) (2017) 83–90.
[12] E. Noether, Invariante variations probleme, Nachr. Akad. Wiss. Gött. Math. Phys. Kl., 2 (1918), 235–257. (English translation in Transp. Theory Stat. Phys. 1 (3) (1971) 186–207.
[13] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1986.
[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 and W. Hereman, The homotopy operator method for symbolic integration by parts and inversion of divergences with applications, Appl. Anal. 87 (2010) 433–455.