Novel computational technique; the second positive member in a new completely integrable hierarchy

Abstract

This paper investigates novel solitary wave solutions of the second positive member in a new utterly integrable hierarchy by employing a novel computational technique. This new method is called the extended Khater (Ext Khat.) method, and it is the first time to apply it to any nonlinear evolution equation. Besides its large number of solutions but it also covers six different analytical schemes’ solutions. Consequently, these methods can be considered as special cases of our new technique. Many distinct solutions are obtained, and no one of them is singular. It is also considered an additional advantage of our new technique, where many of previous employed computational methods get some singular solutions among their solutions. The singular solution does not have any physical meaning, which makes it meaningless. Our new technique is applied here to a Lax representation and bi–Hamiltonian structure. Many solutions are obtained in different formulas such as hyperbolic, trigonometric, rational, and so on.Additionally, some solutions are represented through different sketch styles. Moreover, the stability property of these solutions is also investigated based on the properties of the Hamiltonian system. Finally, our paper’s originality is investigated by comparing our solutions, and those have been obtained in previous work.