New symmetries for a model of fast diffusion

Abstract

In this Letter we prove that, for some partial differential equations that model diffusion, by using the nonclassical method we obtain several new solutions which are not invariant under any Lie group admitted by the equations and consequently which are not obtainable through the classical Lie method. For these partial differential equations that model fast diffusion new classes of symmetries are derived. These nonclassical potential symmetries allow us to increase the number of exact explicit solutions of these nonlinear diffusion equations. These solutions are neither nonclassical solutions of the diffusion equation nor solutions arising from classical potential symmetries. Some of these solutions exhibit an interesting behavior as a shrinking pulse formed out of the interaction of two kinks.