Euler–Poincaré flows and leibniz structure of nonlinear reaction–diffusion type systems

In this paper we present Euler–Poincaré formulation of the Fisher, Fitzhugh–Nagumo, Burgers–Huxley and extended Fitzhugh–Nagumo and extended Burgers–Huxley type nonlinear reaction–diffusion systems. All these flows are related to infinite dimensional almost Poisson manifolds and the corresponding Lie–Poisson structures yield Leibniz brackets, a bracket endowed with both symmetric and skewsymmetric parts. The symmetric part contributes the diffusion part of the ssystem. The properties exhibited by the reaction–diffusion systems defined in this way are in general very different from the standard Hamiltonian mechanics since the dynamics are controlled by the standard Poisson brackets. Moreover, all the nonlinear reaction–diffusion systems under consideration are Euler–Poincaré flows on the dual of Kirillov’s superalgebra associated to the Bott–Virasoro group.