Nonlinear systems of PDEs admitting infinite-dimensional Lie algebras and their connection with Ricci flows

A wide class of two-component evolution systems is constructed admitting an infinite-dimensional Lie algebra. Some examples of such systems that are relevant to reaction–diffusion systems with cross-diffusion are highlighted. It is shown that a nonlinear evolution system related to the Ricci flow on warped product manifold, which has been extensively studied by several authors, follows from the above-mentioned class as a very particular case. The Lie symmetry properties of this system and its natural generalization are identified and a wide range of exact solutions is constructed using the Lie symmetry obtained. Moreover, a special case is identified when the system in question is reducible to the fast diffusion equation in one space dimension. Finally, another class of two-component evolution systems with an infinite-dimensional Lie symmetry that possess essentially different structures is presented.