On Soliton Solutions of the Anti-Self-Dual Yang-Mills Equations from the Perspective of Integrable Systems

Abstract

In this thesis, we construct a class of exact ASDYM 1-solitons and multi-solitons on 4-dimensional real spaces with the Euclidean signature $(+, +, +, +)$, the Minkowski signature $(+, - , -, -)$, and the split signature ($+$, $+$, $-$, $-$) (the Ultrahyperbolic space). They are new results and successful applications of the Darboux transformation introduced by Nimmo, Gilson, Ohta. In particular, the principal peak of the Lagrangian density Tr$F_{\mu\nu}F^{\mu\nu}$ is localized on a 3-dimensional hyperplane in 4 dimensional space. Therefore, we use the term "soliton walls" to distinguish them from the domain walls. For the split signature, we show that the gauge group can be $G=\mathrm{SU}(2)$ and $G=\mathrm{SU}(3)$ and hence the soliton walls could be candidates of physically interesting objects on the Ultrahyperbolic space $\mathbb{U}$. On the other hand, we use the techniques of the quasideterminants to show that in the asymptotic region, the ASDYM $n$-soliton possesses $n$ isolated distributions of Lagrangian densities with phase shifts. Therefore, we can interpret it as $n$ intersecting soliton walls.