International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 3, Pages 593-608
Rearrangement of lattice particles
University of North Dakota, Physics Department, Grand Forks 58201, ND, USA
Received 9 May 1991; Revised 3 October 1991
Copyright © 1993 M. Muraskin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 became regular. In this
paper, we investigate various approaches with the aim of enlarging the nonsymmetric regions.
Integrating in 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).