The study of shock fronts and diffusive soliton waves to energy dissipation along with invariance classification

This study presents a symmetry-based analytical investigation of the Bateman-Burgers equation, a significant nonlinear model describing shock formation in compressible fluids, sonic booms, plasma wave damping, and boundary-layer dynamics. A complete Lie symmetry analysis is performed, yielding a five-dimensional symmetry algebra comprising time and space translations, scaling, and dilation symmetries. The obtained symmetries are employed to reduce the governing partial differential equation to ordinary differential equations through similarity transformations. Subsequently, the generalized logistic method and the Riccati-Bernoulli sub-ODE technique are utilized to construct novel invariant solitary wave solutions. Unlike previous studies, which primarily focused on obtaining isolated wave structures without a symmetry-based classification, the present work establishes a unified Lie-invariant framework that guarantees the uniqueness and invariance properties of the derived solutions. The resulting wave profiles, including solitary and kink-type structures, are visualized through two-dimensional, three-dimensional, contour, and layered graphical representations to examine their propagation characteristics. The analysis reveals rich nonlinear dynamical behavior and provides a systematic classification of solution families that cannot be transformed into one another through symmetry operations. These findings enhance the theoretical understanding of the Bateman-Burgers equation and offer potential applications in nonlinear optics, fluid mechanics, plasma physics, and related areas of wave propagation.