Systems in Chemotaxis: Mathematical Modeling, Invariant Analysis, Solitons and Numerical Solution

This paper investigates a nonlinear chemotaxis model involving diffusion and chemically directed transport. A detailed Lie symmetry analysis is carried out for different parameter cases, and the corresponding determining equations are derived explicitly to classify the admitted Lie point symmetries. Using the obtained symmetry generators, several similarity reductions are constructed, including time-invariant, space-invariant, scaling, and traveling-wave reductions, with the reduced ordinary differential systems derived step by step. In particular, traveling-wave transformations reduce the governing PDE system to ordinary differential equations, from which several exact wave profiles are obtained, including multi-wave, breather-type, and kink-rational interaction solutions. The analytical structure and graphical behavior of these solutions are examined with the term soliton used only when the corresponding localization properties are satisfied. In addition, conservation-law approaches are employed to explore the structural properties of the model. Finally, the method of lines is used to obtain numerical approximations, allowing for a comparison with the analytical profiles and illustrating the influence of model parameters on the cell-density and chemical-concentration dynamics.