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

We study dynamics of multi-soliton solutions of anti-self-dual Yang-Mills equations for G = GL(2, C) in four-dimensional spaces. The one-soliton solution can be interpreted as a codimension-one soliton in four-dimensional spaces because the principal peak of action density localizes on a three-dimensional hyperplane. We call it the soliton wall. We prove that in the asymptotic region, the n-soliton solution possesses n isolated localized lumps of action density, and interpret it as n intersecting soliton walls. More precisely, each action density lump is essentially the same as a soliton wall because it preserves its shape and velocity except for a position shift of principal peak in the scattering process. The position shift results from the nonlinear interactions of the multi-solitons and is called the phase shift. We calculate the phase shift factors explicitly and find that the action densities can be real-valued in three kind of signatures. Finally, we show that the gauge group can be G = SU(2) in the Ultrahyperbolic space U (the split signature (+, +, -)). This implies that the intersecting soliton walls could be realized in all region in N=2 string theories. It is remarkable that quasideterminants dramatically simplify the calculations and proofs.

Automated summary

This study investigates the dynamics of multi-soliton solutions of anti-self-dual Yang-Mills equations for G = GL(2, C) in four-dimensional spaces. It shows that n-soliton solutions have n isolated localized lumps of action density, which can be interpreted as n intersecting soliton walls. The analysis reveals that the action densities can be real-valued in three different signatures. It also suggests that intersecting soliton walls can be realized in all regions of N=2 string theories.