Department of Engineering Mathematics,
University of Bristol.
The Lorentz Center at the University of Leiden, The Netherlands ( http://www.wi.leidenuniv.nl/~vorst/lorentz.html) is currently running a programme in `Dynamical Systems and Pattern Formation' from 15th September to 15th November. The program involves a visitor program and a series of workshops coinciding with Professor Jack Hale's visit as the Kloosterman Professor at the Mathematical Institute in Leiden during the Fall of 1997
The Lorentz Center is a study center, rather like the Newton Institute in Cambridge. specifically aimed at organising longer periods of concentration in astronomy, mathematics, physics and computer science. The Center is housed on the third floor of the modernistic tilting(!) ``Oortgebouw'' building. Its 25 offices, all of them supplied with workstations, common room with exceptionally comfortable sofas, and full secretarial support offer an ideal environment for research collaboration and interaction.
The program on pattern formation has been centred around six 2-4 day workshops organised by different groups on topics of current interest:
1) Finite dimensional dynamical systems, September 29 - October 1, H.W. Broer and G. Vegter.
2) Multi-bump solutions, October 6 - October 9, A. Doelman, L.A. Peletier and R.C.A.M. van der Vorst
3) Dynamics of differential equations with delays, October 13 - October 15, J.K. Hale and S.M. Verduyn-Lunel.
4) Operators and dynamical systems, October 23 - October 25, Ph. Clement, M.A. Kaashoek and S.M. Verduyn-Lunel.
5) Interactions in space: improving the mean field approximation, October 29 - November 1, O. Diekmann and J.A.J. Metz.
6) Interfaces and parabolic regularisation, November 5 - November 7, C.J. van Duijn and J. Hulshof
Several researchers from the Universities of Bath and Bristol attended the meeting on Multi-bump solutions. John Toland, in one of the main evening lectures, presented results on a new variational formulation of the steady water wave problem which was further amplified in a presentation by Boris Buffoni. Alan Champneys presented results on homoclinic orbits to saddle centre equilibria with application in nonlinear optics, and Mark Peleter, although sadly ill for his presentation, was due to talk on new work with Chris Budd and Giles Hunt on a 1D PDE formulation of geological folding.
The meeting brought together an exciting mixture of new methods and problems around the timely topic of the existence and stability of multi-hump homoclinic, heteroclinic and periodic solutions of ODEs and PDEs. The field has undergone rapid progress in recent years and it was stimulating to see the range of different mathematical approaches; e.g. variational methods, shooting, numerics, Evans function approach for stability, singular perturbations, and enhanced Melnikov-type techniques. A recurring theme was problems which are fourth-order in space (or coupled systems of second order problems); e.g. the Extended Fischer Kolmogorov, Swift-Hohenberg and 5th-order KdV equations, and several different problems in nonlinear optics, phase transitions and fluid mechanics.
The environment and the relaxed scheduling of talks meant that genuine interaction took places across methods and applications. The organisers should be congratulated for a meeting that more than fulfilled its potential!
UK participants at other workshops in the programme have included/will include S. van Strien (Warwick) at the first workshop; H. Logemann (Bath) at the meeting on delays; D. Rand & M. Kirkilionis, (Warwick) at the fifth workshop; and Chris Budd & Victor Galaktinov (Bath) and J.R. King (Nottingham) at the final meeting.