Abstract: The non-linear wave equation (sometimes called the non-linear Klien-Gordon equation) is a much studied equation with applications in Josephson transmission lines, dislocations in crystals, DNA processes and much more.  It possess many types of solutions which may include stationary front solutions.  In this talk we shall consider what effect adding a step-like inhomogeneity has on the stability of such fronts and present some of the results we have derived over the last three years.

Abstract: In this talk I will review my previous work on stability and mixing in two-dimensional vortices. This will include a threshold calculation for the existence of nonlinear cat's eye structures and an investigation into nonlinear vorticity staircase structures in a Gaussian vortex.

Abstract: Given a Lie group G acting smoothly on a Riemannian manifold X, let c0 ,c1 : D  → X be smooth functions, where D is also a smooth manifold (possibly with boundary). A geodesic relative to (c0 , c1) is a curve g : D → G with the property that gc0(s) = c1(s) for all sD, which minimises an energy integral over all such curves.

The talk will focus on G=SE(2), X=E2, D=[0,1] and applications to curve-matching. Another case, applicable to matching grayscale images, will also be mentioned if time permits.

Abstract:

Fish appear to swim by periodically moving their fins. Additionally, if they momentarily stop or swim through a tangle of seaweed they appear to recover speed very quickly soon after. This mixture of stability and oscillatory behavior suggests that swimming is a limit cycle. I will provide theoretical evidence which suggests the answer is ``yes''. This will be done by defining a transitive Lie groupoid which can be used for fluid structure interaction. The base of this groupoid will be a set of embeddings of a body into space. We will then define the Lagrangian on the Lie algebroid, and add a time periodic force to the shape of the body as well as a viscous dissipation force due to the non-zero viscosity of the fluid. We will then perform a reduction by SE(3) to obtain a Lie groupoid where the base is the shape space of the swimmer. We will go over some arguments which suggest that a limit cycle could exists in this reduced system. Assuming a limit cycle does exist, we can use reconstruction formulas to obtain the motion of the fish. However, many analytical issues remain to be discussed.

Abstract:

Point vortices are singularities in the vorticity field of a perfect fluid. Physically, they show up as points in the fluid around which the fluid rotates, much like the fluid rotating around the core of a hurricane in the atmosphere.  From a fluid-dynamical point of view, the dynamics of point vortices forms a (finite-dimensional) Hamiltonian system, the solutions of which are weak solutions of Euler's equation, and there is consequently a great deal of interest in modeling and simulating the dynamics of point vortices.

Despite the fact that the equations of motion are Hamiltonian, however, this system has a number of special features which make direct numerical simulation hard: the vector field is not Lipschitz continuous, the symplectic form preserved is not the standard one, and there exists no Lagrangian formulation. The latter two problems especially make the construction of a symplectic integrator for this system a non-trivial task.

In this talk I will use the Hopf fibration to develop a new formulation of the point vortex equations on the sphere S^2. More specifically, I will show that there exists a nonlinear-Schrodinger-like equation on the Lie group SU(2), or alternatively the space of unit quaternions, whose solutions project down onto the trajectories of point vortices in S^2. Using the connection one-form of the Hopf fibration, I will also construct a Lagrangian for these equations, and we will investigate some symplectic integrators based on this formulation. We will illustrate the long-term qualitative features of these integrators with a number of examples from the recent literature, such as the spherical von Karman vortex street.

Abstract:

We give a brief general review of the ADE classification problem. The survey includes simple Kleinian singularities, symmetries of Platonic solids, finite subgroups of SU(2), the Mckay correspondence, integer matrices of norm 2 and Brieskorn’s theory of subregular orbits. We conclude with some joint work with H. Sabourin and P. Vanhaecke on transverse Poisson structures to subregular orbits in semisimple Lie algebras. We show that the structure may be computed by means of a simple Jacobian formula, involving the restriction of the Chevalley invariants on the slice. In addition, using results of Brieskorn and Slodowy, the Poisson structure is reduced to a three dimensional Poisson bracket, intimately related to the simple rational singularity that corresponds to the subregular orbit.  Finally we present some recent results on the minimal orbit.