Nonlinear chains of atoms (NCA) are complex systems with rich dynamics, that influence various scientific disciplines. The lie symmetry approach is considered to analyze the NCA. The Lie symmetry method is a powerful mathematical tool for analyzing and solving differential equations with symmetries, facilitating the reduction of complexity and obtaining solutions. After getting the entire vector field by using the Lie scheme, we find the optimal system of symmetries. We have converted assumed PDE into nonlinear ODE by using the optimal system. The new auxiliary scheme is used to find the Travelling wave solutions, while graphical behaviour visually represents relationships and patterns in data or mathematical models. The multiplier method enables the identification of conservation laws, and fundamental principles in physics that assert certain quantities remain constant over time.

This paper exhibits a brief study of Marangoni convective, Activation energy, non-uniform heat generation, thermal-bioconvection and Oxytactic microorganisms, and time-dependent and stagnation point flow of MHD trihybrid nanofluid around a rotating sphere based on the extended model, referred to as the YOM (Yamada-Ota model) and XM (Xue model). The porous medium impact of Darcy-Forchheimer is included in the momentum equation. Some acceptable transformations are implemented to translate the governing system of PDEs (partial differential equations) and the relevant boundary conditions for computation. This model aims to evaluate the efficiency of two well-known trihybrid nanofluid models, Yamada-Ota and Xue. The shooting approach (BVP4C) is applied to derive the numerical estimates for the relevant differential expression system. The detailed analysis and simulation of trihybrid nanofluid flow patterns are facilitated by the variation in changing factors, which include temperature, velocity, concentration, and gyrotactic and Oxytactic microorganisms the profiles. The results indicate that the YOM performs significantly superior than the XM flow model in terms of heat transmission ability.

Lie symmetry analysis has been applied to the extended Boiti-Leon-Manna-Pempinelli (eBLMP) equation. This system illustrates the exchange of information between two waves with distinct dispersion characteristics. The optimal system of the corresponding Lie algebra has been constructed. The equation considered has been reduced into a simpler form for the computation of analytical solutions. The novelty of this research is the optimal system of subalgebras in one dimension using the adjoint action approach. To analyze and understand the eBLMP more clearly, graphs have been plotted. We have also found conservation laws