The Ward equation, also called the modified 2 + 1 chiral model, is obtained by a dimension reduction and a gauge fixing from the self-dual Yang-Mills field equation on R 2,2 . It has a Lax pair and is an integrable system. Ward constructed solitons whose extended solutions have distinct simple poles. He also used a limiting method to construct 2-solitons whose extended solutions have a double pole. Ioannidou and Zakrzewski, and Anand constructed more soliton solutions whose extended solutions have a double or triple pole. Some of the main results of this paper are: (i) We construct algebraic Bäcklund transformations (BTs) that generate new solutions of the Ward equation from a given one by an algebraic method. (ii) We use an order k limiting method and algebraic BTs to construct explicit Ward solitons, whose extended solutions have arbitrary poles and multiplicities. (iii) We prove that our construction gives all solitons of the Ward equation explicitly and the entries of Ward solitons must be rational functions in x, y and t. (iv) Since stationary Ward solitons are unitons, our method also gives an explicit construction of all k-unitons from finitely many rational maps from C to C n .

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.

No abstract

Abstract. We derive an adjunction inequality for any smooth, closed, connected, oriented 4-manifold X with b + = 1. This inequality depends only on the cohomology algebra and generalizes the inequality of Strle in the case of b1 = 0. We demonstrate that the inequality is especially powerful when 2χ+3σ ≥ 0, whereχ is the modified Euler number taking account of the cup product on H 1 (X; Z).