CONTINUUM MECHANICS AND HAMILTONIAN PDES

This section will focus on REs, RPOs and other aspects of the dynamics of a number of model problems from continuum mechanics. Specific objectives include:

>

Rigorous existence and stability theorems for REs and RPOs of the averaged $\alpha$-Euler equations for statistical properties of inviscid Newtonian fluids and for inviscid non-Newtonian second grade fluids. Analogous treatments of reduced averaged models from elasticity theory. The effects of viscous perturbations on REs and RPOs of these equations.

>

Studies of point vortex models of two-dimensional fluid flows, including:

>

Existence, stability and bifurcations of REs and RPOs;

>

Existence of heteroclinic cycles and associated symmetric chaos;

>

The influence of the non-compact Euclidean symmetry group on the dynamics of planar systems;

>

The effects of viscosity on REs, RPOs and heteroclinic cycles;

>

Embeddings of point vortex system dynamics in $\alpha$-Euler equations.

>

Development of geometric Hamiltonian frameworks for elastic rods and shells and studies of REs, RPOs and dynamics within these frameworks.

>

The stability and bifurcations of REs and RPOs of affine (pseudo) rigid body models of fluid masses, hyperelastic bodies and atomic nuclei. Investigations of heteroclinic chaos in these models. Application of the quantum-classical correspondence principle to affine rigid body systems and comparison with models of atomic nuclei. A study of non-holonomic affine rigid bodies.

>

Stability and bifurcations of pulses in optical fibres, modelled by solitary waves and fronts (REs and RPOs) of (coupled) nonlinear Schrödinger equations. Applications to the phenomenon of `soliton switching'. A study of the effects of non-Hamiltonian perturbations, including dissipation, on these phenomena.

Major breakthroughs are expected to include the development of new techniques which will greatly simplify the computation of stability properties and bifurcations in infinite dimensional continuum systems and the discovery of new dynamical phenomena in models of continuum systems.