IMI Colloquium

The 1969 paper by Bluman and Cole stimulated a lot of research on invariant solutions that could not be recovered from Lie’s classical method. Only a special class of reaction-diffusion equations has full nonclassical reductions to solutions that are not invariant under classical symmetries. However some of those equations have important applications.

For 1+1 dimensional linear diffusion with a nonlinear reaction, only equations such as the Fitzhugh Nagumo equation, and the Huxley equation, with cubic source terms, have strictly nonclassical invariant solutions. Under the assumptions set down by Fisher in 1930, the advance of a new advantageous gene through a diploid population, is governed not by Fisher’s equation but by Huxley’s equation.


For nonlinear reaction-diffusion equations in two or three spatial dimensions, there is a single restriction relating nonlinear diffusivity to nonlinear reaction, that always allows nonclassical reduction to the linear Helmholtz equation. This allows us to construct a class of unsteady solutions to a reaction-diffusion equation with Arrhenius reaction term, that follows from the Gibbs non-analytic temperature-dependent probability distribution.


By analogous means, this also allows for an exact solution of a realistic model for water infiltration into a soil with moisture-dependent plant-root extraction. When the reaction term is a Fisher logistic growth term of a harvested population, there is a critical size of protection zone above which the species is safe from extinction. For the cubic Huxley growth term of population genetics, new advantageous genes are never safe from elimination by harvesting at the boundary of the protection zone.