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The Kawahara equation (KE) and the modified Kawahara equation (MKE) are important modeling equations to interpret shallow water waves with surface tension, magneto-acoustic wave in plasma and gravity waves. In this study, the stable broad-ranging soliton solutions including the well-known bell-shape soliton, anti-bell shape soliton, periodic soliton, compacton, kink, etc. are established through the sine-Gordon expansion (SGE) method. The effect of the physical parameters, namely the dispersion and perturbed dispersive coefficients in the surface elevation are shown through the figure. The surface elevations remain ascertainable for diverse values of the physical and the associated free parameters. The 3D and contour plot of the ascertained solutions clarifies the surface wave properties. The achieved results for each modeling equation provide a significant contribution in analyzing the ion-acoustic waves in plasma, gravity waves, the surface waves in shallow water. This study asserts that the SGE method is reliable, competent, easy and powerful for extracting closed form soliton solutions.

Stable soliton solutions to the shallow water waves and ion-acoustic waves in a plasma