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

Abstract

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 TrFμνFμν 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.

Soliton solutions of noncommutative anti-self-dual Yang–Mills equations

“…We note that a simple parameter choice of one soliton solutions in the non-Wronskian type yields trivial energy densities (See section 4 in[23]) while the present Wronskian-type solitons are non-trivial.…”

“…In this subsection, we focus on one-soliton solutions for J ∈ U (2). In commutative spaces, they are obtained and energy densities of them are real-valued [23]. A noncommutative candidate is given by:…”

“…Gauge fields are calculated as in (4.10) from this J and field strengths are obrtained. In the calculations, the star-products always disappear due to the property (5.6) and the energy density is reduced to the commutative one [23]:…”

Multi-soliton dynamics of anti-self-dual gauge fields

“…Therefore, the description of tau-functions remains to be clarified. Furthermore, several attempts were made by us to construct one-soliton solutions from [2], and the resulting action density is TrF µν F µν = 0 [19]. Perhaps for this reason, only few discussions have been made (as far as the authors know) in this direction for a long time (e.g.…”

“…This Wronskian type solutions can be represented in terms of quasideterminants [9] (called the quasi-Wronskian solutions, for short). A highly nontrivial result of quasi-Wronskian solutions is that the action density in one-soliton case is no longer zero [19]. Moreover, the principal peak of the action density lies on a three-dimensional hyperplane.…”