An investigation of two integro-differential KP hierarchy equations to find out closed form solitons in mathematical physics

Nonlinear partial differential equations (NLPDEs) are widely utilized in engineering and physical research to represent many physical processes of naturalistic occurrences. In this paper, we investigate two well-known NLPDEs, namely, the (2 + 1)-dimensional first integro-differential KP hierarchy equation and the (2 + 1)-dimensional second integro-differential KP hierarchy equation, through a well-stable algorithm known as the (G′G′+G+A)-expansion approach for the first time. This algorithm is generally based on the expansion of function method and has the advantage of easy implementation and can provide a reliable solution to any NLPDEs. Employing the algorithm, we have been able to perceive the closed form solitons of the two chosen NLPDEs that physically represent the solitary wave solutions like, singular, singular periodic, bell, and anti-bell-shaped types of solitons. Furthermore, we explore the graphical manifestations of the obtained solutions, which are of the mentioned soliton types. From the findings of our in-depth study, we can state that the acquired solutions for the selected two equations may greatly aid to extracting the associated natural phenomena in mathematical physics such as fluid dynamics and ocean engineering.