Research Interests

Most of my past and current research projects are concerned with understanding the formation of patterns and the dynamics of nonlinear waves in spatially extended systems. Extended systems are typically modelled by partial differential equations on unbounded domains. Nonlinear waves correspond to interfaces, or defects, that are formed between co-existing patterns. These patterns, as well as the defects and interfaces formed between them, are found in many biological, chemical, and physical applications. Examples are the transmission of signals in optical fibers, the formation of hexagonal and stripe patterns in fluid convection, and the generation of spiral waves in catalytic chemical reactions. Motivated by experiments and numerical simulations, I aim to understand when and how patterns and defects are formed, how they behave under small perturbations, what other patterns or waves with a more complicated spatio-temporal behaviour can bifurcate from them, and how they interact with each other or with domain boundaries. To answer these questions, I use a mixture of analytical and geometric dynamical-systems techniques, and I have also developed numerical algorithms for the computation of waves and their bifurcations.

Lecture notes, tutorials, and links to presentations about my research projects can be found under Online notes.

Overview of recent projects

Undergraduate Research Experiences

Summer@ICERM 2016

I have worked with several groups of undergraduate students on research projects. Example projects and summaries of outcomes can be found on the undergraduate research page. During the summer 2016, Margaret Beck, Todd Kapitula, and I co-organized Summer@ICERM: a group of 26 undergraduate students (including 8 Brown students) worked in 8 teams on projects on lead propagation, voting and zebrafish models, equation-free modelling, snaking bifurcations, and Koopman theory. The summaries of the research teams give an overview of the results obtained during the summer.

Planar spot patterns

localized spots

Localized spot patterns occur in many physical processes. Examples are vegetation patches, coherent light pulses in lasers, and chemical waves. Time-periodic spots (referred to as oscillons) have also been observed in vertically shaken granular media and clay suspensions. Somewhat surprisingly, not much is known about the bifurcations at which planar spots emerge. Our goal is to investigate common bifurcations such as Turing, Hopf, Turing-Hopf and their temporally forced analogues to see whether they can lead to spots. One outcome of our analyses is that spots bifurcate at Turing bifurcations. Unexpectedly, the precise conditions depend on the underlying space dimension: planar spots exist in regions where 1D spots are not possible.

Lattice systems

exchange lemma

Many techniques exist to investigate pulses and fronts of partial differential equations in one space dimension. Far less is known for travelling waves in systems that are posed on lattices, which often provide more appropriate models (eg when considering nerve impulse propagation in myelinated fibers). The FitzHugh-Nagumo equation exemplifies this situation: much is known for its continuous version due to geometric singular perturbation theory, while only numerical results are available for its discrete analogue. Recent work in my group resulted in extensions of geometric singular perturbation theory (and specifically the Exchange Lemma) to ill-posed functional differential equations of mixed type that govern the existence of travelling waves in lattice systems. In particular, we showed that the discrete FitzHugh-Nagumo equation admits stable fast pulses.

Mode-locked lasers

waveguide array

Passively mode-locked lasers generate high-power coherent light pulses through a combination of nonlinear gain and intensity discrimination in a passive laser cavity. Intensity discrimination has been achieved, for instance, by placing semiconductor saturable absorbers in a cavity or, more recently, by coupling the cavity to a waveguide array. In collaboration with Nathan Kutz's group, we investigate these mechanisms with the aim of optimizing output peak power and energy with respect to physical parameters in the laser cavity.

Nonlinear stability of sources

sources

While nonlinear stability of localized patterns is often easily inferred from their spectral stability, the situation is quite different for defects that are embedded in a spatially periodic background. Sources are active defects that select the wavenumber of the periodic background: examples are 1D sources and spiral waves that occur in many biological and chemical reactions (eg in the catalysis of CO on platinum surfaces and in cardiac tissue). Using pointwise Green's function techniques, we investigated the nonlinear stability of time-periodic Lax shocks in viscous conservation laws and of semidiscrete Lax shocks that arise in numerical approximations. Our goal is to extend these techniques to 1D sources.

Localized planar hexagon and roll patches

hexagon patch
worm patch

Localized hexagon and worm patches have been observed in reaction-diffusion systems, optical systems, electrical discharges, liquid crystals, and even in ferrofluids. These structures often exhibit snaking: our numerical computations show that, for each systems parameter value in the snaking region, an infinite number of patterns co-exist that are connected in parameter space and whose width increases without bound. The precise way in which the pattern profiles change along the bifurcation branch is complicated and not well understood. For essentially one-dimensional patterns (for instance, patterns that are localized in one direction and periodic in the others), we have developed theoretical approaches that explain the structure of the bifurcation diagrams and allows us to predict all asymmetric patterns if we know all symmetric patterns.