A bistable reaction-diffusion system in a stretching flow

We examine the evolution of a bistable reaction in a one-dimensional stretching flow, as a model for chaotic advection. We derive two reduced systems of ordinary differential equations (ODEs) for the dynamics of the governing advection-reaction-diffusion partial differential equations (PDE), for pulse-like and for plateau-like solutions, based on a non-perturbative approach. This reduction allows us to study the dynamics in two cases: first, close to a saddle-node bifurcation at which a pair of nontrivial steady states are born as the dimensionless reaction rate (Damkoehler number) is increased, and, second, for large Damkoehler number, far away from the bifurcation. The main aim is to investigate the initial-value problem and to determine when an initial condition subject to chaotic stirring will decay to zero and when it will give rise to a nonzero final state. Comparisons with full PDE simulations show that the reduced pulse model accurately predicts the threshold amplitude for a pulse initial condition to give rise to a nontrivial final steady state, and that the reduced plateau model gives an accurate picture of the dynamics of the system at large Damkoehler number. Published in Physica D (2006)

Asymptotic analysis of combined breather-kink modes in a Fermi-Pasta-Ulam chain

We find approximations to travelling breather solutions of the one-dimensional Fermi-Pasta-Ulam (FPU) lattice. Both bright breather and dark breather solutions are found. We find that the existence of localised (bright) solutions depends upon the coefficients of cubic and quartic terms of the potential energy, generalising an earlier inequality derived by James [CR Acad Sci Paris 332, 581, (2001)]. We use the method of multiple scales to reduce the equations of motion for the lattice to a nonlinear Schr{\"o}dinger equation at leading order and hence construct an asymptotic form for the breather. We show that in the absence of a cubic potential energy term, the lattice supports combined breathing-kink waveforms. The amplitude of breathing-kinks can be arbitrarily small, as opposed to traditional monotone kinks, which have a nonzero minimum amplitude in such systems. We also present numerical simulations of the lattice, verifying the shape and velocity of the travelling waveforms, and confirming the long-lived nature of all such modes.