### EXACT SOLUTION AND CONSERVED VECTORS OF FIFTH-ORDER KORTEWEG-DE VRIES EQUATION OBTAINED BY LIE SYMMETRY ANALYSIS

#### Abstract

We investigate point symmetries of the Fifth-order Korteweg-de Vries (KdV) equation by applying Lie theory of continuous groups, from which we get two lists of infinitesimal generators forming two Lie algebras, and the latter will turn out to be a sub-space of the former. A general solution of seven arbitrary parameters satisfying these Lie symmetries is constructed. A Lagrangian is found for the simultaneous system of the KdV equation and its adjoint equation, which is used to study the conserved vectors of the system via Noether’s theorem. Lengthy computations in obtaining the extended infinitesimal generator, solving the system of PDEs, and applying Euler-Lagrange operator were handled by PDEtools and DifferentialGeometry Maple packages.

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PDF#### References

Brauer, K. (2000). The Korteweg-de Vries equation: history, exact solutions, and graphical representation. University of Osnabrück/Germany1.

Eriksson, M. (2008). Symmetries and Conservation Laws Obtained by Lie Group Analysis for Certain Physical Systems (Doctoral dissertation, Thesis. Uppsala School of Engineering/Uppsala University, 2008. Web).

Ibragimov, N. H. (2007). A new conservation theorem. Journal of Mathematical Analysis and Applications, 333(1), 311-328.

Bluman, G., & Anco, S. (2008). Symmetry and integration methods for differential equations (Vol. 154). Springer Science & Business Media.

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**MAYFEB Journal of Mathematics**

**Toronto, Ontario, Canada**

**MAYFEB TECHNOLOGY DEVELOPMENT**

**ISSN**

**2371-6193**