Nonlinear Systems of PDEs Admitting Infinite-Dimensional Lie Algebras and Their Connection With Ricci Flows. II: The Two-Dimensional Space Case

Abstract

Motivated by previous results in special cases associated with Ricci flows, all possible two-components evolutions systems of (1+2)-dimensional second-order partial differential equations (PDEs) admitting an infinite-dimensional Lie algebra are constructed. It is shown that a natural generalization of this Lie algebra to the higher-dimensional case does not lead to a more general result because the infinite-dimensional symmetry is broken. The recently derived system, which is related to Ricci flows, is identified as a very particular case among the evolution systems obtained. All possible radially symmetric stationary solutions of the Ricci-flow-associated special case are then constructed using the surprisingly rich Lie algebra of the resulting reduced system of ordinary differential equations (ODEs), exemplifying the exceptional status of such systems. Moreover, it is proved that this Lie algebra is reducible to the fifteen-dimensional algebra of the simplest system of two second-order ODEs. Several time-dependent exact solutions in the radially symmetric case are constructed as well. It is shown that the solutions obtained are bounded and smooth provided arbitrary parameters are correctly specified. By their nature, geometric PDEs typically enjoy rich symmetry properties; our analysis illustrates how those properties may be extrapolated to broader classes of models that are of independent interest.