4.1
The application of techniques from bifurcation theory and geometry to
the fundamental models of continuum mechanics has proved to be very
successful in the stability analysis of REs and attendant bifurcation
phenomena.
These methods, combined with new techniques from section 1, will be applied
to a number of continuum problems, including the 
-Euler
equations. In the last two years it has shown that
they are much better behaved than the usual Euler equations in terms of both
the existence and uniqueness of solutions and non-zero viscosity
perturbations. They therefore provide an excellent context for developing
and testing
rigorous stability, bifurcation and non-Hamiltonian perturbation
theories for REs and RPOs of continuum systems.
The same techniques will be used to derive and study
analogous equations from nonlinear elasticity theory. Special emphasis will
be put on new models for shape memory alloys for which the fundamental
equations have not yet been studied mathematically.
4.2 There is a huge literature, stretching back over a century, on the derivation and dynamics of systems of point vortices. However it is only in the relatively recent past that geometric and symmetry techniques have been applied systematically, resulting in existence, stability and bifurcation theorems for REs, descriptions of dynamics for small numbers of vortices and computations of geometric phases. In this project we will extend the results for REs, develop analogous results for RPOs, investigate non-zero visocity perturbations and use systems of point vortices as test-beds for exploring and testing new ideas and techniques for symmetric chaos.
4.3 The derivation of equations for elastic rod and shell dynamics as limits of 3-D elastic body equations has been approached from several different points of view including asymptotic analysis and Galerkin methods. However only in the past few years has a start been made on a fully Hamiltonian perturbation theory for these and related problems and there have been very few applications of this theory to the stability and dynamics of rod and shells. A study of REs and RPOs within the context of this theory is one of the aims of this project.
4.4 The history of affine (or pseudo) rigid body models of elastic and fluid bodies goes back to the work of Dedekind, Dirichlet and Riemann on self-gravitating fluid masses. Recent work has included RE stability calculations for potentials coming from elasticity theory and Nekhoroshev stability theory for REs. The parallel quantum theory has also been studied, motivated by the possibility that affine rigid bodies may serve as a useful model for atomic nuclei. The network programme includes investigations of: (a) how the global structure of the set of REs depends on the potential energy function; (b) bifurcations of RPOs; (c) the existence of heteroclinic cycles and symmetric chaos; (d) quantum manifestations of classical phenomena; (e)non-holonomic affine rigid bodies.
4.5 Linear instability results for solitons and front solutions in evolutionary Hamiltonian wave equations have been obtained recently by introducing the multi-symplectic Evans function. This project will look at the influence of non-Hamiltonian perturbations on such solutions. It will also look at the application of these theories to pulses in optical fibres, especially the so called `soliton switching' effect in coupled fibres.