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].

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