In this article, two different fifth-order nonlinear evolution equations (NLEEs), namely the (2+1)-dimensional bidirectional Sawada-Kotera (bSK) and the (2+1)-dimensional variable-coefficient Sawada-Kotera (SK) equations are investigated, whose generous exact solutions are formally generated, respectively. The paper proceeds step-by-step with increasing detail about derivation processes, first illustrating the algorithms of the proposed approach and then exploiting an even deeper connection between the derived solitary solutions with the assumed solution form. After applying a special trial function along with the related Hirota bilinear equations to handle these two cases a variety of traveling wave solutions are extracted, naturally including solitary wave and periodic wave ones. Moreover, via specifying values to the free parameters the physical explanations of the extracted solutions are depicted in detail. Particularly, the results indicate that free parameters can drastically influence the existence of solitary waves, their nature, profile, and stability. They are applicable to enrich the dynamical behavior of the associated nonlinear wave models. Finally, it is shown that the presented methodized resolution is straightforward and applicable to solve other NLEEs.
