Lie symmetries, bifurcation analysis and soliton solutions of the viscous-capillarity compressible fluid model

The dynamics of compressible fluids with viscosity and capillarity remain only partially understood, particularly in relation to phase transitions and nonlinear wave propagation. Due to the difficulties posed by its mixed-type nature, where eigenvalue variations result in hyperbolic-elliptic transitions and shock formation, thorough symmetry-based and bifurcation analyses of the one-dimensional viscous-capillarity compressible van der Waals system (the p-system) remain limited despite prior research. This paper examines the model using soliton solutions, bifurcation theory, and Lie symmetry techniques. Symmetry analysis identifies invariant structures, making analytical and numerical treatment easier. Soliton solutions reveal robust nonlinear waveforms that accurately model shock structures and capillary-driven interfacial phenomena, while bifurcation analysis reveals critical stability thresholds controlled by viscosity and surface tension. In the natural sciences and engineering, where phase transitions and interfacial effects are crucial, the results offer practical significance and a deeper theoretical understanding of nonlinear compressible flows.