Lie Group Analysis for a Multispecies, Spatially Inhomogeneous, Mutually Interacting Gas Mixture

We consider systems of general nonlinear PDEs that arise in the extended kinetic theory of gases describing mixtures of spatially inhomogeneous, mutually interacting species. For general, autonomous, space-independent interaction laws, these systems are always invariant under the translation group. Via Lie group analysis methods, we show that for a Lotka-Volterra interaction law and three or more species, the system is invariant under a three-parameter group of transformations representing scaling, and translation in time and space. Moreover, the Lie group analysis shows that this is the only invariance group of the system.