Magnetic monopoles, or the topological soliton solutions of Yang-Mills-Higgs gauge theories in three space dimensions, have been objects of fascination for over a quarter of a century. BPS monopoles in particular have been the focus of much research. Many striking results are now known, yet, disappointingly, explicit solutions are rather few. We bring techniques from the study of finite dimensional integrable systems to bear upon the construction. The transcendental constraints of Hitchin may be replaced by (also transcendental) constraints on the period matrix. For a class of curves we show how to these may be reduced to a number theoretic problem. A recently proven result of Ramanujan related to the hypergeometric function enables us to solve these and construct the corresponding monopoles.
The Faddeev-Hopf model is a phenomenological model of (among other things) quark confinement, which possesses knot-like topological solitons. Its energy functional is a sum of two terms, one of which is very well understood by geometers (it's the Dirichlet, harmonic map or sigma model energy). The other, while equally geometrically natural, has not been systematically studied. In this talk I will describe the variational calculus for this second functional, obtaining results on the existence and stability of critical points. The results have potential significance for the existence of instantons in the strong coupling limit of the Faddeev-Hopf model.
(Joint work with Martin Svensson, Leeds)
Moutard-Darboux Transformations for general 2D second order differential equation will be presented. Some basic reductions and specifications of the general transformation will be shown. Discretizations of the equations admitting Moutard-Darboux transformation will be discussed.
The connection between integrable partial difference equations and Yang-Baxter maps is explained and some integrable partial difference equations on a 2-dim grid as coupled Yang-Baxter maps are presented. Also higher dimensional analogues of these coupled maps are discussed.
We introduce a general Hamiltonian describing coherent superpositions of Cooper pairs and condensed molecular bosons, which has two integrable sets of parameters. We show how the spectrum for one of the integrable manifolds maps onto quasi-exactly solvable (QES) model with PT-symmetric boundary conditions, whereas the spectrum of the other manifold maps onto a QES model with standard Hermitian boundary conditions.
The integrability of systems of hydrodynamic type by the generalized hodograph method requires the vanishing of the corresponding Haantjes tensor. We generalize this approach to hydrodynamic chains --- infinite-component systems of hydrodynamic type for which the corresponding infinite matrix is `sufficiently sparse'. For such systems the Haantjes tensor is well-defined, and the calculation of its components involves finite summations only. We illustrate our approach by classifying broad classes of conservative and Hamiltonian hydrodynamic chains with the zero Haantjes tensor.
We prove that the vanishing of the Haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi-Hamiltonian hydrodynamic reductions, thus providing an easy-to-verify necessary condition for the integrability.
The path integral approach to the quantization of one degree-of-freedom Newtonian particles is considered within the discrete time-slicing approach, as in Richard Feynman's original development. It is well known that in the time-slicing approximation the quantum mechanical evolution will generally not have any stationary states. In the development of this talk we look for conditions on the potential energy term such that the quantum mechanical evolution may possess stationary states -- without having to perform a continuum limit. In the classical limit the systems thus obtained are integrable maps.
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Last updated 28th February, 2005.