New improvement of the ϕ6-model expansion method and its applications to the new (3+1)-dimensional integrable Kadomtsev–Petviashvili equation

In this paper, an improvement for the ${\varphi }^6$-model expansion method is presented. In this approach, contrary to the classical ${\varphi }^6$-model expansion method, obtaining explicit solutions for nonlinear ordinary and partial differential equations is congenial and undemanding of any constraint conditions, where the method can be applied and used for obtaining solutions without having any conditions on them. Moreover, the new approach is used to obtain new solutions for the new $(3+1)$-dimensional integrable Kadomtsev–Petviashvili equation. We demonstrated that for the same equation, the classical ${\varphi }^6$-model expansion and the improved ${\varphi }^6$-model expansion approaches produce the same family of solutions. However, the improved ${\varphi }^6$-model expansion method is found to be more efficient and convenient.