The Ξ -expansion method for several types of nonlinear PDEs and traveling wave solutions.

In this article, we applied the improved and generalized method to some nonlinear problems such as: the (2+1) dimensional Bogoyavliskii equation and the Ablowitz-Kuap-Newell-Segur equation (AKNSE), but distinctively and simple, we called it the Ξ -expansion method. This method is characterized by the addition of an arbitrary constant d to the general solution, which leads to the presentation of new exact solutions. This method is simple but powerful for extracting exact solutions of partial differential equation, which habitually occur in mathematical physics, engineering sciences and applied mathematics. [ABSTRACT FROM AUTHOR]