Soliton solutions with stability, bifurcation analysis and phase portraits of Kudryashov–Sinelshchikov equation

This paper investigates the Kudryashov–Sinelshchikov equation, which describes nonlinear pressure wave propagation in liquid-gas bubble mixtures while incorporating the effects of viscosity and heat transfer. By employing a suitable traveling-wave transformation, the governing partial differential equation is reduced to an ordinary differential equation and analyzed within the approach of dynamical systems. The resulting singular system is regularized through an appropriate transformation of the independent variable, which preserves its first integral and enables detailed phase-portrait analysis. The qualitative behavior of the system is further explored using phase-plane diagrams and bifurcation analysis to identify critical transitions in wave dynamics. To construct explicit analytical solutions, the Hirota bilinear method is applied, leading to exact one-soliton, two-soliton, and three-soliton solutions. These solutions are graphically demonstrated through two-dimensional surface plots, three-dimensional visualizations, and contour profiles, which clearly illustrate the localized, stable, and interaction-preserving characteristics of the solitons. The results emphasize not only the integrability of the Kudryashov–Sinelshchikov model but also its capability to capture intricate nonlinear wave dynamics of significant physical relevance. By combining dynamical systems analysis with exact multi-soliton solutions, this study provides deeper insights into the behavior and interactions of pressure waves in dispersive and thermoviscous environments. Such an enhanced understanding establishes a valuable theoretical approach for modeling nonlinear phenomena in liquid-gas mixtures, where viscosity and heat transfer play a central role.