Ansatz-based exploration of M-shaped and multi-wave solitons in the Kudryashov–Sinelshchikov model

In this research, we utilized a symbolic computation framework that combines ansatz functions with logarithmic transformations to derive various rational solutions for the Kudryashov–Sinelshchikov (KS) equation. This nonlinear model captures the dynamics of pressure waves in liquid-gas bubble mixtures while incorporating the effects of heat transfer and viscosity. By employing the three-wave transformation technique, we unveiled the formation of multi-wave solitons and analyzed the properties of homoclinic breathers using exponential and cosine ansatz functions. The quadratic ansatz further revealed the emergence of M-shaped rational solitons. Additionally, we investigated the complex interactions between kink waves and M-shaped solitons under two distinct scenarios, identifying novel structures like kink-cross-rational solitons through quadratic, cosine, and hyperbolic transformations. To highlight the physical relevance of these findings, we adopted an extensive visualization approach, including 2D and 3D representations, contour diagrams, and stream plots, accompanied by a comprehensive stability analysis and detailed tabulated results. Our research advances previous efforts by providing a more in-depth understanding of the interplay of nonlinearity, dissipation, and wave transformations, paving the door for possible applications in engineering and physical sciences. These innovative solutions offer significant potential applications in areas such as plasma physics, optical fiber technology, and other advanced scientific domains.