## Research Interests

### Soliton dynamics

There are many systems of partial differential equations of interest in
theoretical physics which possess soliton solutions (stable, smooth,
localized lumps of energy). Often the space of * static *
soliton solutions forms a smooth manifold, called the moduli space,
which can be equipped with a
natural Riemannian metric. This is true, for example, of the
Yang-Mills-Higgs, abelian Higgs and nonlinear sigma models, whose solitons
are called respectively monopoles, vortices and lumps. Following a
conjecture of N.S. Manton, it is believed that slow soliton dynamics in such
systems is well approximated by geodesic motion in the corresponding
moduli space (in fact this has been proved for monopoles and vortices by
D. Stuart). This leads one to study the Riemannian geometry of soliton
moduli spaces and exploit this geometry to understand slow soliton dynamics.
My own research concentrates on a particular sigma model called the CP1
model. Here the moduli space is the space of holomorphic maps from a given
compact Riemann surface to CP1. In particular I have studied in detail the
dynamics of 1 lump moving on a sphere
[3],
[13],
[16],
2 lumps
moving on a torus
[6]
and (with I.A.B. Strachan)
1 lump moving under the
influence of its own gravitational effects
[9]. I have
also proved (with L.A. Sadun) that the moduli space is always geodesically
incomplete
[7].
Most recently I have made (with M. Haskins)
some progress towards adapting Stuart's
analysis for vortices and monopoles to the case of sigma models
[17].

In the context of gauge theory, I have developed a point particle
formalism to understand the interactions of well-separated Landau-Ginzburg
vortices, yielding a simple model of the scattering of type II vortices
[5] and
(with N.S. Manton)
critically coupled vortices
[18]. In the critically
coupled case, we obtain a formula for the asymptotic metric on the N-vortex
moduli space, which has many nice geometric properties.
The point particle formalism can also be used to analyze the first order
Hamiltonian dynamics of vortices in thin superconductors, as shown in joint
work with Nuno Romão [19].
### Discrete solitons

In applications in condensed matter and biophysics, solitons usually
propagate through discrete spaces (crystal lattices for example), and it
has long been recognized that spatial discreteness introduces crucial and
highly complex phenomena into the soliton dynamics. My research in this area
follows three strands:

(1) The study of non-standard discretizations which preserve "topological"
features of the soliton dynamics
[1],
[4] and
[10].

(2) Semi-classical quantum dynamics of solitons on lattices
[2],
[11].

(3) The study of breathers (time periodic oscillatory solitons) in discrete
systems
[8],
[12],
[14],
[15].

[ University of Leeds ]
[ Mathematics ]
[ Pure ]
[ Applied ]
[ Statistics ]