Theoretical and numerical investigations on solitary wave solutions of Gardner equation

Abstract

This paper formulates new hyperbolic functions ansatze to construct exact solitary wave solutions of the Gardner (combined KdV-mKdV) equation and a finite element approach for the numerical solutions. A novel class of exact solitary wave solutions is derived. The conditions on the physical parameters for the existence of the obtained structures are also presented. Accuracy of the proposed numerical scheme is assessed in terms of L2 and $L_{\infty}$ error norms. Numerical experiments demonstrate the accuracy and robustness of the method which can be further used for solving other nonlinear problems.