American Journal of Mathematical and Computational Sciences 2020; 5(2): 9-16 http://www.aascit.org/journal/ajmcs Lie Symmetries and Invariant-Solutions of the Potential Korteweg-De Vries Equation Faya Doumbo Kamano 1 , Bakary Manga 2 , Joel Tossa 3 , Momo Bangoura 4 1 Department of Research, Distance Learning High Institute, Conakry, Guinea 2 Department of Mathematics and Computer Sciences, University of Cheikh Anta Diop, Dakar, Senegal 3 Institute of Mathematics and Physical Sciences, University of Abomey-Calavi, Porto-Novo, Benin 4 Department of Mathematics, University of Gamal Abdel Nasser, Conakry, Guinea Email address * Corresponding author Citation Faya Doumbo Kamano, Bakary Manga, Joel Tossa, Momo Bangoura. Lie Symmetries and Invariant-Solutions of the Potential Korteweg-De Vries Equation. American Journal of Mathematical and Computational Sciences. Vol. 5, No. 2, 2020, pp. 9-16. Received: April 17, 2020; Accepted: June 6, 2020; Published: August 5, 2020 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 1. Introduction Lie symmetries of differential equations are one of the important concepts in the theory of differential equations and physics. Among others methods, Lie method is a firm one for finding symmetries of differential equations. This method was first applied to determine point symmetries (see [8] and [12]). In 1969-1970, B. Kent Harrison and Frank Estabrook devised a method to calculate symmetries of differential equations using differential forms and Cartan’s formulation of differential equations [2]. They were simply trying to understand how the symmetries of Maxwell’s equations could be found from the differential form version of those equations. Once they realized that the key to symmetries was the use of the Lie derivative, B. Kent Harrison applied the method to several others equations such as the one dimensional heat equation, the Short wave gas dynamic equation and the nonlinear Poisson equation (see [3] and [4]). Here we apply this method to the potential KdV equation, given as follows   + (  )  +  =0, (1) where u(x,t) is a function of space x and time variable t; subscripts denoted partial derivatives; a is a real constant, with a is no vanishing. The p-KdV equation is widely used in various branches of physics [1]. Exact travelling wave solutions to nonlinear evolution equations, particularly which appear in many physical structures in solitary wave theory such as solitons, kinks, peakons, and cuspons [14], draw considerable interest