Techniques, exact and approximate, for solving the anti-self-dual Yang-Mills equations

This work, on solving the anti-self-dual Yang-Mills (ASDYM) equations, can be broken down into two major parts, the first of which involves developing new mathematical techniques to be applied to the various formulations of the ASDYM equations, while the second involves finding explicit solutions to the, ASDYM equations. The first of the mathematical techniques is the construction of the spin-weighted Green's functions for the edh operator and powers of the edh operator. This work was extended to include the construction of Green's functions for the edh-bar operator and powers of the edh-bar operator as well as a procedure for obtaining the Green's functions for any combination of products of the edh and edh-bar operators. These Green's functions are used to solve the Sparling equation for the ASDYM equations (or any other spin-weighted differential equation involving the edh and/or edh-bar operators). Another of the mathematical techniques is solving the matrix-valued Riemann-Hilbert problem for the case of triangular data. This result is an extension of a previously obtained solution for the case of 2 x 2 triangular data and is applied to the twistor construction of the ASDYM equations. On the problem of finding explicit solutions to the ASDYM equations, the major accomplishment of this work is the development of an approximation and triangularization procedure for the twistor construction. A general patching matrix P (representing the data) for the twistor construction ill approximated (this approximation being equivalent to the Ward Ansaetze). The approximate patching matrix P(m) is triangularized and the associated Riemann-Hilbert problem solved, thereby generating an anti-self-dual solution of the Yang-Mills equations. The approximate patching matrices and the associated (exact) anti-self-dual Yang-Mills solutions are then shown to form a dense subset in the entire solution space