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The Sawada-Kotera equations illustrate the non-linear wave phenomena in shallow water, ion-acoustic waves in plasmas, fluid dynamics, etc. In this article, the two-mode Sawada-Kotera equation (tmSKE) occurring in fluid dynamics is considered which is important model equations for shallow water waves, the capillary waves, the waves of foam density, the electro-hydro-dynamical model. The improved F-expansion and generalized exp(−φ(ζ))-expansion methods are utilized in this model and abundant of solitary wave solutions of different kinds such as bright and dark solitons, multi-peak soliton, breather type waves, periodic solutions, and other wave results are obtained. These achieved novel solitary and other wave results have significant applications in fluid dynamics, applied sciences and engineering. By granting appropriate values to parameters, the structures of few results are presented in which many structures are novel. The graphical moments of the results are provided to signify the impact of the parameters. To explain the novelty between the present results and the previously attained results, a comparative study has been carried out. The restricted conditions are also added on solutions to avoid singularities. Furthermore, the executed techniques can be employed for further studies to explain the realistic phenomena arising in fluid dynamics correlated with any physical and engineering problems.INDEX TERMS Improve F-expansion method; Generalized exp(−φ(ζ))-expansion method; Two-mode Sawada-Kotera equation; Traveling and dual wave solutions, breather waves, Periodic solitons.

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