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

Abstract

AbstractA 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.