On the space-time Monopole equation

Abstract

The space-time monopole equation is obtained from a dimension reduction of the anti-self dual Yang-Mills equation on ${\mathbb{R}}^{2,2}$. 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 Bäcklund transformations to construct a large class of monopoles which are global in time and have both continuous and discrete scattering data.

31 pages

Cited by in corpus (4)

• Moduli-Space Dynamics of Noncommutative Abelian Sigma-Model Solitons
• Scattering of Noncommutative Waves and Solitons in a Supersymmetric Chiral Model in 2+1 Dimensions
• The Cauchy problem of the Ward equation with mixed scattering data
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