Generalized symmetries and higher order conservation laws of the camassa-Holm equation

  • Parastoo Kabi-Nejad Mathematics Department, Iran University of Science and Technology, Tehran, Iran
  • Mehdi Nadjafikhah Mathematics Department, Iran University of Science and Technology, Tehran, Iran
Keywords: Camassa-Holm equation, Generalized Symmetry, Optimal System, Conservation law


In the present paper, we derive generalized symmetries of order three of the Camassa–Holm equation. In addition, one–dimensional optimal system of Lie subalgebras are investigated. Furthermore, the 2-dimensional homotopy formula is employed to construct higher–order conservation laws for the Camassa–Holm equation.


Download data is not yet available.


Bila, N. (1999). Infinitesimal symmetries of Camassa-Holm equation. Paper presented at the Proceedings of the Conference of Geometry and its Applications in Technology and Workshop on Global Analysis, Differential Geometry and Lie Algebras, BSG Proceeding 4, 1999.

Bluman, G. W., Cheviakov, A. F., & Anco, S. C. (2010). Applications of symmetry methods to partial differential equations (Vol. 168): Springer.

Camassa, R., & Holm, D. D. (1993). An integrable shallow water equation with peaked solitons. Physical review letters, 71(11), 1661.

Camassa, R., Holm, D. D., & Hyman, J. M. (1994). A new integrable shallow water equation Advances in Applied Mechanics (Vol. 31, pp. 1-33): Elsevier.

Clarkson, P., Mansfield, E. L., & Priestley, T. (1997). Symmetries of a class of nonlinear third-order partial differential equations. Mathematical and Computer Modelling, 25(8-9), 195-212.

Constantin, A. (2001). On the scattering problem for the Camassa-Holm equation. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 457(2008), 953-970.

Constantin, A., & Escher, J. (1998). Wave breaking for nonlinear nonlocal shallow water equations. Acta Mathematica, 181(2), 229-243.

Fuchssteiner, B., & Fokas, A. S. (1981). Symplectic structures, their Bäcklund transformations and hereditary symmetries. Physica D: Nonlinear Phenomena, 4(1), 47-66.

Hereman, W. (2006). Symbolic computation of conservation laws of nonlinear partial differential equations in multi‐dimensions. International journal of quantum chemistry, 106(1), 278-299.

Hereman, W., Colagrosso, M., Sayers, R., Ringler, A., Deconinck, B., Nivala, M., & Hickman, M. (2005). Continuous and discrete homotopy operators and the computation of conservation laws Differential Equations with Symbolic Computation (pp. 255-290): Springer.

Holm, D. D., Marsden, J. E., & Ratiu, T. S. (1998). The Euler–Poincaré equations and semidirect products with applications to continuum theories. Advances in Mathematics, 137(1), 1-81.

Kara, A., & Bokhari, A. H. (2011). A non-variational approach to the construction of new ‘higher-order’conservation laws of the family of nonlinear equations α (ut+ 3uux)+ β (utxx+ 2uxuxx+ uuxxx)− γuxxx= 0. Communications in Nonlinear Science and Numerical Simulation, 16(11), 4183-4188.

Kouranbaeva, S. (1999). The Camassa–Holm equation as a geodesic flow on the diffeomorphism group. Journal of Mathematical Physics, 40(2), 857-868.

Misiołek, G. (1998). A shallow water equation as a geodesic flow on the Bott-Virasoro group. Journal of Geometry and Physics, 24(3), 203-208.

Naz, R., Naeem, I., & Abelman, S. (2009). Conservation laws for Camassa–Holm equation, Dullin–Gottwald–Holm equation and generalized Dullin–Gottwald–Holm equation. Nonlinear Analysis: Real World Applications, 10(6), 3466-3471.

Olver, P. J. (2000). Applications of Lie groups to differential equations (Vol. 107): Springer Science & Business Media.

How to Cite
Kabi-Nejad, P., & Nadjafikhah, M. (2019). Generalized symmetries and higher order conservation laws of the camassa-Holm equation. International Journal of Fundamental Physical Sciences (IJFPS), 9(2), 20-25.