On the space-time Monopole equation

Details On the space-time Monopole equation On the space-time Monopole equation

2006-02-27

arxiv.org/abs/math/0602607v1

Mathematics

Differential Geometry

31 pages

Scientific paper

The space-time monopole equation is obtained from a dimension reduction of the anti-self dual Yang-Mills equation on $\R^{2,2}$. A family of Ward equations is obtained by gauge fixing from the monopole equation. In this paper, we give an introduction and a survey of the space-time monopole equation. Included are alternative explanations of results of Ward, Fokas-Ioannidou, Villarroel and Zakhorov-Mikhailov. The equations are formulated in terms of a number of equivalent Lax pairs; we make use of the natural Lorentz action on the Lax pairs and frames. A new Hamiltonian formulation for the Ward equations is introduced. We outline both scattering and inverse scattering theory and use B\"acklund transformations to construct a large class of monopoles which are global in time and have both continuous and discrete scattering data.

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