abstract: This talk will concentrate on how asymptotic techniques may be adapted to study discrete problems, such as difference, differential-difference, or lattice equations. I will demonstrate examples in which asymptotic methods such as multiple scales and exponential asymptotics are applied to difference and differential-difference equations.
I will consider the following illustrative examples:
- Using multiple scales to capture delayed bifurcations in the slowly-varying discrete logistic equation (work with C. Hall). I will show that the most useful approach to this problem is to consider discrete fast variation, with a slow time scale treated as continuous. This method requires the use of matched asymptotic expansions in order to combine the time scales correctly. - Finding generalised solitary waves in a diatomic Toda lattices (work with M. Porter). I will show that exponential asymptotic methods can be applied directly to this singularly perturbed system in order to study solitary waves through such systems, and show that they produce non-decaying oscillations in the wake of the solitary wave. I will also show that there is a simply asymptotic formula for mass ratios which eliminate the non-decaying oscillations, producing classical solitary waves.
If time permits, I will briefly summarise results obtained by applying exponential asymptotics to a range of other discrete equations, including discrete KdV and Painleve equations (with N. Joshi and S. Luu), the advance-delay NLS equation (with D. Pelinovsky), and the biharmonic NLS equation (with M. Porter).