Self-Dual Yang-Mills Fields in Static Axially Symmetric Case and Related Topics

In this article we will present an explicit geometric picture about the complete integrability of the static axially symmetric SDYM equation and the gravitational Ernst equation, interpret the correspondence between their Bäcklund transformation formulae and the transformations from one focal surface of Weingarten congruence to the other, and give the matrix Riccati equation so that the integrability of the B.T. will be proved. It is shown that for the axially symmetric SDYM equation and gravitational Ernst equation the adjoint space of the group (SL(2r)) is a 3-dimensional Minkowski space, and the corresponding soliton surfaces have negative variable curvature. After introducing the generator R we can explain the B.T. as the rotation around the common tangent between two surfaces of solitons. Using Riccati equation we will confirm in this paper the integrability of B.T. and prove that the B.T. is strong, i.e., the new and old solutions satisfy equations of motion separately. Some related topics are also discussed.