Painlevé properties, gauge invariance and solitons of some classes of reaction-diffusion equations

The role of symmetries and conservation laws are well known mechanisms for the reduction of systems of differential equations and, used in conjunction, lead to double reductions of the underlying models. We show here that variational and gauge symmetries have additional applications in the integrability of differential equations. In particular, we present a broad class of diffusion type equations, viz., the Fisher–Kolmorov and Fitzhugh–Nagumo equations, that satisfy Pinlevé properties, of their respective travelling wave forms and solitons, under the existence of gauge symmetries following a variational principle.