The (2 + 1)-dimensional Chaffee-Infante equation and the dimensionless form of the Zakharov equation have widespread scopes of function in science and engineering fields, such as in nonlinear fiber optics, the waves of electromagnetic field, plasma physics, the signal processing through optical fibers, fluid dynamics, coastal engineering and remarkable to model of the ion-acoustic waves in plasma, the sound waves. In this article, the first integral method has been assigned to search closed form solitary wave solutions to the previously proposed nonlinear evolution equations (NLEEs). We have constructed abundant soliton solutions and discussed the physical significance of the obtained solutions of its definite values of the included parameters through depicting figures and interpreted the physical phenomena. It has been shown that the first integral method is powerful, convenient, straightforward and provides further general wave solutions to diverse NLEEs in mathematical physics.