Numerical Simulation of Kink Collisions, Analytical Solutions and Conservation Laws of the Potential Korteweg–de Vries Equation

Abstract

In this study, we investigate the nonlinear potential Korteweg–de Vries equation (pKdVe) by making use of the Lie group analysis. We start by constructing Lie symmetries and thereafter utilize them to execute symmetry reductions of pKdVe. We then obtain solutions of the pKdVe by using the direct integration method. The obtained solutions are demonstrated in respect of Jacobi elliptic functions. Some of the obtained solutions are illustrated graphically. Moreover, we obtain four conserved vectors of the pKdVe by making use of the multiplier method and five conserved vectors by using the theorem owing to Ibragimov. Finally, we simulate collisions between kinks for the pKdVe.