This article aims to study the exact solitary wave solutions of some Nonlinear Evolution Equations (NLEEs) by using a novel expansion method. Three nonlinear models of physical significance i.e. Pochhammer-Chree Equations (PCE) equation, Newell-Whitehead Equation(NWE) and Fitzhugh-Nagumo Equation(FNE) are considered. The method that we have chosen, is simple, straightforward and, gives the three types of solutions including trigonometric, exponential, and rational solutions as compared to other existing methods. Various types of periodic and solitary wave solutions are derived. Keywords: Solitons, NLEEs, Newell-Whitehead Equation(NWE), Soliton, Fitzhugh-Nagumo Equation(FNE), Pochhammer-Chree Equations (PCE).