Nottingham

Whispering gallery modes in optical resonators have received a lot of attention as a mechanism for constructing small, directional lasers. They are also potentially important as passive optical components in schemes for coupling and filtering signals in optical fibres, in sensing devices and in other applications. In this talk it is argued that the evanescent field outside resonators that are very slightly deformed from circular or spherical is surprising in a couple of respects. First, even very small deformations seem to be capable of leading to highly directional emission patterns. Second, even though the undelying ray families are very regular and hardly differ from the integrable circular or spherical limit inside the resonator, a calculation of the evanescent field outside it is not straightforward.

This is because even very slight nonintegrability has a profound effect on the complexified ray families which guide the external wave to asymptopia. An approach to describing the emitted wave is described which is based on canonical perturbation theory applied to the ray families and extended to comeplx phase space.

I will provide an overview of theoretical models aimed at understanding how self-excited oscillations arise when flow is driven through a finite-length flexible tube or channel. This problem is approached using a hierarchy of models, from one to three spatial dimensions, combining both computational and asymptotic techniques. I will explain how recent work is starting to shed light on the relationship between local and global instabilities, energy balances and the role of intrinsic hydrodynamic instabilities. This is collaborative work with Peter Stewart, Robert Whittaker, Jonathan Boyle, Matthias Heil and Sarah Waters.

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 I will discuss the dynamics of synaptically coupled model neurons that undergo a form of accommodation in the presence of sustained activity. The basic model is an integral equation for synaptic activity that depends upon the non-local network connectivity, synaptic response, and firing rate of a single neuron. A phenomenological model of accommodation is examined whereby the firing rate is taken to be a simple state-dependent threshold function. As in the case without threshold accommodation classical Mexican-Hat connectivity is shown to allow for the existence of spatially localised states (bumps). Importantly an analysis of bump stability (in both one and two spatial dimensions) using recent Evans function techniques shows that bumps may undergo instabilities leading to the emergence of both breathers and travelling waves. Numerical simulations show that bifurcations in this model have the same generic properties as those seen in many other dissipative systems that support localised structures, and in particular those of coupled cubic complex Ginzburg-Landau equations, and three component reaction diffusion equations. Interestingly, travelling pulses in this model truly have a discrete character in the sense that they scatter as auto-solitons. /notices/events/abstracts/differential-equations/ht06/Coombes.shtml