The Hamiltonian formalism in reaction-diffusion systems

Abstract

It is well known that the Hamiltonian formalism plays an central role in classical mechanics. In this survey, we show that the Hamiltonian formalism is useful for studying pattern formation problems in reaction-diffusion systems. Although they are not derived from the Newton principle (the motion law), the notion of gradient/skew-gradient structure enables us to use the Hamiltonian formalism for their study. This structure was originally introduced by [14], and formulated in more abstract fashion by [6]. We explain usefulness of the gradient/skew-gradient structure through the linear stability analysis of standing pulse solutions and spatially periodic patterns in reaction-diffusion systems of activator-inhibitor type.