New exact solutions for the loaded Korteweg-de Vries and the loaded modified Korteweg-de Vries by the functional variable method

In this paper, we construct exact traveling wave solutions of the loaded Korteweg-de Vries and the loaded modified Korteweg-de Vries by the functional variable method. The performance of this method is reliable and effective and gives the exact solitary and periodic wave solutions. All solutions to these equations have been examined and 3D graphics of the obtained solutions have been drawn using the MATLAB program. We get some traveling wave solutions, which are expressed by the hyperbolic functions and trigonometric functions. The graphical representations of some obtained solutions are demonstrated to better understand their physical features, including bell-shaped solitary wave solutions, singular soliton solutions, and solitary wave solutions of kink type. Our results reveal that the method is a very effective and straightforward way offormulating the exact traveling wave solutions of non-linear wave equations arising in mathematical physics and engineering.