Determination of the rich structural wave dynamic solutions to the Caudrey–Dodd–Gibbon equation and the Lax equation

This article addresses the implementation of the new generalized $$\left( {G^{\prime } /G} \right)$$ -expansion method to the Caudrey–Dodd–Gibbon (CDG) equation and the Lax equation which are associated with the fifth-order KdV (fKdV) equation. The method works well to derive a variety of standard and functional closed-form wave solutions with distinct physical structures, such as, soliton, kink, periodic soliton, and bell-shaped soliton solutions. The solutions obtained using this method are useful and adequate than other methods. In order to understand the physical aspects and importance of the method, the attained solutions have been simulated graphically. The extracted results definitely establish that the new generalized $$\left( {G^{\prime } /G} \right)$$ -expansion method is an effective mathematical tool to work out new solutions to different types of local nonlinear evolution equations emerging in applied science and engineering, but this method is not effective in solving nonlocal equations.