New soliton solutions of anti-self-dual Yang-Mills equations

We study exact soliton solutions of anti-self-dual Yang-Mills equations for $G =GL(2)$ in four-dimensional spaces with the Euclidean, Minkowski and Ultrahyperbolic signatures and construct special kinds of one-soliton solutions whose action density Tr$F_{\mu\nu}F^{\mu\nu}$ can be real-valued. These solitons are shown to be new type of domain walls in four dimension by explicit calculation of the real-valued action density. Our results are successful applications of the Darboux transformation developed by Nimmo, Gilson and Ohta. More surprisingly, integration of these action densities over the four-dimensional spaces are suggested to be not infinity but zero. Furthermore, whether gauge group $G= U(2)$ can be realized on our solition solutions or not is also discussed on each real space.
Comment: 19 pages; Dedicated to the memory of Jon Nimmo; v2: minor changes, discussion on singularities added, version to appear in JHEP