Renormalization Group and the Melnikov Problem for PDE's

We give a new proof of persistence of quasi-periodic, low dimensional elliptic tori in infinite dimensional systems. The proof is based on a renormalization group iteration that was developed recently in [BGK] to address the standard KAM problem, namely, persistence of invariant tori of maximal dimension in finite dimensional, near integrable systems. Our result covers situations in which the so called normal frequencies are multiple. In particular, it provides a new proof of the existence of small-amplitude, quasi-periodic solutions of nonlinear wave equations with periodic boundary conditions.Comment: 44 pages, plain Te

Asymptotics of solutions in nA+nB->C reaction Diffusion systems

We analyze the long time behavior of initial value problems that model a process where particles of type A and B diffuse in some substratum and react according to $nA+nB\to C$. The case n=1 has been studied before; it presents nontrivial behavior on the reactive scale only. In this paper we discuss in detail the cases $n>3$, and prove that they show nontrivial behavior on the reactive and the diffusive length scale.Comment: 22 pages, 1 figur

Asymptotics of solutions in an A + B → C reaction-diffusion system

We analyze the long time behavior of an initial value problem that models a chemical reaction-diffusion process A + B → C. The problem has previously been studied by Gálfi and Rácz, who predicted the critical indices associated with the reaction by using a scaling ansatz motivated by numerical simulations. In this paper we point out some difficulties which appear in problems of this type due to the non-uniform convergence of the solution towards the scaling limit, and solve them by giving an explicit description of the corrections scaling that have to be concluded to prove bounds on the solution that are uniform in space and time. This allows us to relate rigorously the critical exponents as computed from the scaling ansatz to the exponents of the reaction