Rearrangement of lattice particles - M. Muraskin (International Journal of Mathematics and Mathematical Sciences)
We have studied origin point data which lead to soliton loop lattice systems when we specify an integration path in no integrability Aesthetic Field Theory. When we applied the integration scheme developed in previous paper we found that the solitons get rearranged. Close to the origin we saw a system more disorderly than the lattice. However, farther from the origin in two dimensional maps the location of planar maxima (minima) for fixed y became regular. In this paper, we investigate various approaches with the aim of enlarging the nonsymmetric regions. Integrating in z did not lead to an enlarged nonsymmetric region. We were able to enlarge the region by altering the magnitudes appearing in the origin point data. It is not clear if we can continually enlarge the nonsymmetric region by this method. We studied what we call an imperfect lattice which in a coarse sense can be thought of as being comprised of soliton loops when we specify an integration path. Here the integration scheme did not lead to an exact symmetry, but there was a repeat of type structures (as indicated by observations of contour lines in the maps). We then extended the system to higher dimensions. In particular, we studied a complex six dimensional space which is a natural extension of Minkowski space as an example. The system studied gave rise to a loop lattice, but with magnitudes of maxima (minima) of the different loops varying in an oscillatory way. When we applied the integration scheme to this system found no sign of the previously discussed symmetry in the domain studied although the system is not free from other regularities (this is also the case when magnitudes are altered).