The Hamiltonian formalism in reaction-diffusionsystems

摘要

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