On the space-time monopole equation

Abstract

Abstract. 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 Bäcklund transformations to construct a large class of monopoles which are global in time and have both continuous and discrete scattering data.

Local well-posedness for the space–time Monopole equation in Lorenz gauge

“…4 Note that Theorem 1.1 follows immediately from Theorem 2.1. To prove Theorem 2.1 we will first diagonalise the left-hand side of (2.1).…”

Global solutions to space–time monopole equations in one space dimension

“…It was first introduced in [9] as a hyperbolic analog of Bogomolny equations and discussed from the point of view of twistors. A broad survey on the space-time monopole system is given in [4]. In particular, using the inverse scattering transform, they have shown global existence and uniqueness up to a gauge transformation for small initial data in W 2,1 (R 2 ).…”

Almost critical local well-posedness for the space-time monopole equation in Lorenz gauge

“…The space-time Monopole equation was first introduced by Ward [9] as a space-time analog of Bogomolny Equations or Magnetic Monopole equations. For more detailed survey of the equation see [6].…”