Dynamics of two-mode Sawada-Kotera equation: Mathematical and graphical analysis of its dual-wave solutions

New dual-wave soliton solutions are addressed for the two-mode Sawada-Kotera (TmSK) equation arising in fluids by the modified Kudryashov and new auxiliary equation methods. As outcomes, bright, dark, periodic, and singular-periodic dual-wave solutions are obtained. The graphs of the solutions are provided to show the impact of the parameters. A comparison between our solutions and the existing ones is carried out.