Workshop on Difference Equations
University of Adelaide
9 April 1999
Location of talks: "the Loft", 2nd level, Mitchell building
9.00 a.m. Colin Rogers (UNSW)
9.50 a.m. Michael Murray (Adelaide)
10.40 a.m. Frank Nijhoff (Leeds)
11.30--11.40 am Morning Tea
11.40 a.m Reinout Quispel (Latrobe)
12.30 p.m. Andy Hone (Adelaide)
1.00--2.00 p.m. Lunch
2.00 p.m. Wolfgang Schief (UNSW)
2.50 p.m. David Adams (Adelaide)
3.40--3.50 p.m. Afternoon Tea
3.50 p.m. Nail Akhmediev (ANU)
4.40 p.m. Sasha Kitaev (Adelaide&Steklov)
5.30 p.m. Sam Yates (Adelaide)
6.00 p.m. End of Workshop

Colin Rogers:
Backlund Transformations and the Discretisation of Characteristic Systems
Abstract: Links between Backlund transformations and the discretisation of characteristic systems for Monge-Ampere equations are discussed along with applications in gasdynamics and nonlinear elasticity.

Michael Murray:
On the complete integrability of the discrete Nahm equations
Abstract: The discrete Nahm equations, a system of matrix valued difference equations, arose in the work of Braam and Austin on half-integral mass hyperbolic monopoles.
We show that the discrete Nahm equations are completely integrable in a natural sense: to any solution we can associate a spectral curve and a holomorphic line-bundle over the spectral curve, such that the discrete-time DN evolution corresponds to walking in the Jacobian of the spectral curve in a straight line through the line-bundle with steps of a fixed size. Some of the implications for hyperbolic monopoles are also discussed. This talk is based on the paper math-ph/9903017.
Frank Nijhoff:
Lattice structures associated with the sixth Painleve equation
Abstract: TBA

Reinout Quispel:
Geometric numerical integration methods for ordinary differential equations
Abstract:Traditional numerical integration methods (such as Runge-Kutta methods) are all-purpose methods,ie they work for all ordinary differential equations (ODEs).Recently it has become clear that for special classes of ODEs (eg for Hamiltonian ODEs) special novel classes of integration methods should be used.We will present a classification of these novel "geometric" integration methods.This classification is based on classical work by Lie and Cartan on infinite-dimensional Lie groups.

Andy Hone:
The exact discrete Ermakov-Pinney equation

Abstract: By exploiting the connection with Schrodinger operators and the Darboux transformation, a Backlund transformation for the Ermakov-Pinney (EP) equation is obtained. This leads to an exact discrete EP equation, as well as a second discrete version, in a natural way. The connection with discrete Schrodinger equations and discrete Schwarzians is also derived.

Wolfgang Schief:
Trapezoidal discrete surfaces. Geometry and Integrability
Abstract: Surfaces on which the lines of curvature form geodesics and parallels are discretized in a purely geometric manner. Discrete principal curvatures are defined and it is shown that the natural discrete Gauss equation is given by the standard discrete Schroedinger equation with the discrete Gaussian curvature as its potential. The subclass of discrete surfaces of revolution is considered and used to establish algebraic and geometric properties which are reminiscent of those known in the continuous case. Important connections with integrable difference equations are also recorded.

David Adams:
Degeneracy in discretisations of geometric 1st order differential operators
Abstract: Discrete versions of geometric 1st order differential operators (e.g. the Dirac operator) tend to exhibit a degeneracy feature: the nullspace of the discrete operator has a higher dimension than the nullspace of the corresponding continuum operator. This leads to problems in the discrete formulations of of field theories in physics involving these operators, the simplest and best-known example being the "lattice fermion doubling problem". I will discuss examples of this degeneracy feature, and attempt to explain it as a reflection of the problems that arise when one attempts to construct a discrete analogue of the Hodge star operator. Moreover, I will show how a doubled version of the operator can be discretised in a degeneracy-free way in a lattice-dual lattice framework.

Nail Akhmediev:
Wonders of incoherent solitons: dynamics, collisions and other interactions.
Abstract: New exact solutions show unusual properties of incoherent solitons. Their intensity profile is arbitrary. Moreover, it changes after collisions. They may propagate on a plane wave background. This list may be continued... Detailed comparison of incoherent solitons with properties of fundamental solitons is given.

Alexander Kitaev:
Discretizations of coalescence limits

Abstract: TBA

Sam Yates:
Discrete Morse Theory and extended $L^2$ homology
Abstract: TBA
7 April 1999, Nalini Joshi.