Integrability and Vesture for Harmonic Maps into Symmetric Spaces

After giving the most general formulation to date of the notion of integrability for axially symmetric harmonic maps from R^3 into symmetric spaces, we give a complete and rigorous proof that, subject to some mild restrictions on the target, all such maps are integrable. Furthermore, we prove that a variant of the inverse scattering method, called vesture (dressing) can always be used to generate new solutions for the harmonic map equations starting from any given solution. In particular we show that the problem of finding N-solitonic harmonic maps into a noncompact Grassmann manifold SU(p,q)/S(U(p) x U(q)) is completely reducible via the vesture (dressing) method to a problem in linear algebra which we prove is solvable in general. We illustrate this method by explicitly computing a 1-solitonic harmonic map for the two cases (p = 1, q = 1) and (p = 2, q = 1); and we show that the family of solutions obtained in each case contains respectively the Kerr family of solutions to the Einstein vacuum equations, and the Kerr-Newman family of solutions to the Einstein-Maxwell equations.