On The Monopole Equation And The Ward Equation

The space-time monopole equation is obtained by a dimension reduction from the self-dual (or anti-self-dual) Yang-Mills equation on R2'2. By imposing a gauge fixing from the space-time monopole equation, one can derive the Ward equation or Ward model, any pure soliton solution of which was shown by Ward [57] corresponding to certain holomorphic vector bundle over the compactification space S2 from minitwistor space TP1. In [35], a noncommutative extension of the Ward model was derived from the string theory and later a supersymmetric extension of this noncommutative model was established [33]. These four objects are what we investigate in this dissertation, and all necessary preliminaries and notations are reviewed in Chapter 1.In Chapter 2, we deal with the Cauchy problem for the space-time monopole equation. We give the condition that a spacial pair (i.e., initial data) has con-tinuous scattering data in terms of holomorphic vector bundles. Using the terminology of holomorphic vector bundles and transversality of certain maps, parametrized by initial data, we describe those initial data, with which we can use scattering and inverse scattering method to solve the Cauchy problem up to gauge transformation.In Chapter 3, we explicitly obtain the holomorphic vector bundle corre-sponding to any SU(2) Ward soliton with simple pole data by formulating the meromorphic framing, which is determined by the extended Ward solitons with simple poles. We also give bundles corresponding to several SU(2) Ward 2-solitons with a double pole by the same method and point out some information about the bundles corresponding to SU(2) Ward solitons with general pole data.Chapter 4 is devoted to constructing a large family of multi-soliton solu-tions of the noncommutative extension of the U(n) Ward model. We give a non-commutative version of algebraic Backlund transformation (BT), and use this transformation to construct multi-soliton solutions with simple pole data of this noncommutative model. We extend the limiting method and generalized alge- braic BT [9] to the noncommutative setup, then apply these noncommutatively extended method to construct multi-soliton solutions with arbitrary pole data.In Chapter 5, we construct a large family of multi-soliton solutions of the supersymmetric extension of the noncommutative U(n) Ward model. We present a supersymmetric extension of our noncommutative version of algebraic BT, by which we construct multi-soliton solutions with simple pole data, and of our noncommutative version of limiting method and generalized algebraic BT to construct multi-soliton solutions with general pole data.