# Nonlinear Dynamics and Asymptotics

Dynamical systems, nonlinear waves, asymptotic analysis

### Exponential Asymptotics

Usually asymptotic expansions are divergent, and that means that exponentially small phenomena are hidden in the tail of the divergent series. In exponential asymptotics we make these exponentially small contributions visible. In this way we obtain much more accurate approximations and increase the regions of validity, we obtain the most powerful method to compute the so-called Stokes multipliers, and in many applications explain (and compute) the appearance of oscillatory behaviour when certain Stokes lines are crossed. Topics available are exponential asymptotics for linear or nonlinear ODEs and PDEs with a small parameter. Informal enquiries can be made to Adri Olde Daalhuis (a.oldedaalhuis@ed.ac.uk).

### Uniform Asymptotics for Difference Equations

This field of research is relatively new. For differential equations and integrals it is well-known how to obtain uniform asymptotic expansions. These expansions are valid in large regions, and especially near critical points where the behaviour of the solutions changes dramatically. There are some results in the literature for difference equations, but none of them is as simple or as powerful as what is known for differential equations and integrals. Hence, any new result is worth publishing. Informal enquiries can be made to Adri Olde Daalhuis (a.oldedaalhuis@ed.ac.uk).

### Mixed-mode dynamics

Mixed-mode dynamics is a type of complex oscillatory behaviour that is characterised by an alternation of oscillations of small and large amplitudes. Mixed-mode oscillations (MMOs) frequently occur in fast-slow ('multi-scale') systems of ordinary differential equations; they are, for instance, found in models from mathematical neuroscience, where they correspond to complicated firing patterns seen in experimental recordings of neural activity.Various mechanisms have been proposed to explain mixed-mode dynamics; however, the relationship between them has not been investigated systematically yet. Moreover, relevant neurological models are typically too high-dimensional to be amenable to mathematical analysis, and have to be reduced efficiently to lower-dimensional normal forms which still capture the essential model dynamics. Informal enquiries can be made to Nikola Popovic (Nikola.Popovic@ed.ac.uk).

### Cut-off reaction-diffusion systems

Reaction-diffusion equations, such as the Fisher or the Nagumo equations, have found widespread use as 'minimal' models in the sciences. Solutions that retain a fixed profile in time and space, known as propagating fronts, are frequently relevant as asymptotic ('limiting') states to which general solutions converge. While reaction-diffusion models have been widely applied in the continuum limit of discrete (many-particle) systems, their utility is often limited due to stochastic effects which need to be considered for finite particle numbers. These effects can be approximated by introducing a 'cut-off' function which deactivates the reaction terms whenever the particle concentration lies below some threshold.The impact of such a cut-off approximation has been studied in detail in a number of propagation regimes; however, an in-depth understanding of the stability and convergence of the resulting front solutions is lacking, as is the investigation of cut-off systems in more than one spatial dimension. Informal enquiries can be made to Nikola Popovic (Nikola.Popovic@ed.ac.uk).

### Undular bores in nonlinear optical media

Undular bores are a familiar wave form in fluid mechanics, an example being the tidal bore observed in regions of strong diurnal tides, the Severn Estuary being a famous example. Undular bores can also form from optical beams in nonlinear optical media such as colloids and liquid crystals. There is next to no existing theory to describe undular bores in such media, but there has been considerable experimental interest.This project would involve investigating undular bores in nonlinear optical media.Informal enquiries can be made to Noel Smyth (n.smyth@ed.ac.uk).

### Optical vortices

Optical vortices, the optical equivalent of the familiar fluid vortex, have found widespread use in many fields where they are used to optically trap small objects such as cells. This project would investigate the interaction of optical vortices and the interaction of optical vortices with refractive index changes. The idea behind this is to control the position of optical vortices by optical methods. Informal enquiries can be made to Noel Smyth (n.smyth@ed.ac.uk).

### Bistability of solitary waves in nematic liquid crystals

Optical solitary waves can form in nematic liquid crystals and thermal optical media. An open question of great interest both theoretically and experimentally is whether there is bistability of these solitary waves, that is whether different stable solitary waves with the same power can exist. There is some theoretical indication that this is possible, but the question requires an in depth analysis. Informal enquiries can be made to Noel Smyth (n.smyth@ed.ac.uk).