Exploring the truncated M-fractional exact solitons, modulation instability, and stability analysis of the Kudryashov–Sinelshchikov model

In this paper, various types of exact soliton solutions of the truncated M-fractional Kudryashov–Sinelshchikov equation, a significant fluid surge model, were obtained. This model accounted for density and heat transfer effects while describing the propagation of pressure waves in mixtures of liquid–gas bubbles. By applying the modified $(G^{\prime }/G^2)$-expansion method and the extended Sinh–Gordon equation expansion method, we derived new solutions in the forms of trigonometric, hyperbolic, and rational functions. The obtained solutions were illustrated dynamically using 2D, 3D, and contour plots. These results were novel due to the use of a new definition of fractional derivatives. The effect of the fractional derivative on the solutions was demonstrated through 2D plots. To examine the stability of the obtained solutions, stability analysis was performed. Steady-state solutions were derived using modulation instability analysis. The obtained solutions may be useful in various fields of science and engineering.