American Journal of Mathematical and Computational Sciences  
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Lie Symmetries and Invariant-Solutions of the Potential Korteweg-De Vries Equation
American Journal of Mathematical and Computational Sciences
Vol.5 , No. 2, Publication Date: Aug. 5, 2020, Page: 9-16
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Authors
 
[1]    

Faya Doumbo Kamano, Department of Research, Distance Learning High Institute, Conakry, Guinea.

[2]    

Bakary Manga, Department of Mathematics and Computer Sciences, University of Cheikh Anta Diop, Dakar, Senegal.

[3]    

Joel Tossa, Institute of Mathematics and Physical Sciences, University of Abomey-Calavi, Porto-Novo, Benin.

[4]    

Momo Bangoura, Department of Mathematics, University of Gamal Abdel Nasser, Conakry, Guinea.

 
Abstract
 

The purpose of this paper is to investigate the nonlinear partial differential equation, known as potential Korteweg-de Vries (p-KdV) equation. We have implemented the Harrison technique that makes use of differential forms and Lie derivatives as tools to find the point symmetry algebra for the p-KdV equation. This approach allows us to obtain five infinitesimal generators of point symmetries. Fixing each generator of symmetries that we have found, we construct a complete set of functionally independent invariants, corresponding to the new independent and dependent variables. Using these new variables, called “similarity variables”, the reduced equations have been constructed systematically, which leads to exact solutions that are group-invariant solutions for the p-KdV equation. The obtained solutions are of two types. The reduced equations from the generator of space and time translation groups are the first and the third order ordinary differential equations respectively and lead to the Travelling-invariant solutions. Then, the reduced equation from the generator of the Galilean boosts is the first order ordinary differential equation and leads to Galilean-invariant solutions. Under the generator of scaling symmetries, the potential KdV equation reduces to the third order ordinary differential equation, which does not admit symmetries. And then, there are no functionally independent invariants for that last equation, its solutions are essentially new functions not expressible in terms of standard special functions.


Keywords
 

Symmetries, Differential Forms, Lie Derivative, Korteweg-de Vries Equations, Invariant Solutions


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