Analytical methods for reaction-diffusion equations: Critical wave speeds and axi-symmetric phenomena

This thesis concerns three problems associated with traveling waves in reaction-diffusion equations and one associated with rotating kinematic chains.;First, we study traveling waves in a class of reaction-diffusion equations for the family of potentials fm(U) = 2Um(1 - U) for m ≥ 1. We use perturbation methods, matched asymptotics, multiple scales and exponential asymptotics in both the limits, m → 2 and m → infinity to derive expansions for the critical wave speed, cmax(m), that separates traveling waves with algebraic and exponential tails. The optimum truncation of these expansions is also discussed. Finally, an integral formulation shows that nonuniform convergence of the generalized equal area rule occurs at the critical wave speed. These asymptotic results are significant because the problem is equivalent to a global bifurcation problem, for which analytical methods are presently not available.;Secondly, we analyze a general class of reaction-diffusion equations, Ut = Uxx + f(U), where f(U) is a bistable cubic-like function with f(0) = f( 1) = 0. We focus on deriving new bounds for c max. The classical bound, 2supUe0,1 f U/U , is reviewed, with an emphasis on the triangular trapping region into which the unstable manifold goes. We show how to modify the boundary of the trapping region to obtain sharper estimates.;The third project focuses on axi-symmetric traveling wave solutions of multi-dimensional reaction-diffusion equations using asymptotic analysis and dynamical systems theory. The non-autonomous radial differential equations are expressed as systems with slowly varying phase planes. Analytical results for these phase plane systems are then used to produce the asymptotic forms and speeds of the curved fronts.;Finally, we study the bifurcations and stability of relative equilibria in rotating, n-link kinematic chains suspended in a gravitational field. Viewing the constant speed of the forced rotation as a bifurcation parameter, we find that the bifurcation values of the chain's vertical equilibrium are roots of a certain polynomial. Finally, we establish a relation between the planar chain model and the continuous heavy string.