Study of different lattice solutions in aesthetic field theory

Abstract

We study various types of lattice solutions in the no integrability aesthetic field theory. The field equations collapse into simple systems of equations for the various lattices under study. We display these equations. The lattice systems under study are categorized by the number of field components needed to describe the lattice. Due to the difficulty of finding all the nonlinear relations for the field components, we consider only linear restrictions on the field components in the categorization. The simplest lattice system found is described by two components. We study also a three component and a five component lattice. We investigate various properties of the different lattices. Except for the simplest two component lattice, we find that the summation over paths method leads to planar multi maxima and minima. For the two component lattice we find no planar maxima and minima for the region close to the origin. We study a three dimensional five component lattice. We follow the trajectory of lattice particles on prescribing an integration path at the outset. We find that the motion of lattice particles is in closed loops. The loops arise from what appears to be simple harmonic motion in both the x and y directions. The lattice particles in this system are seen to display soliton behavior. The aesthetic field equations involve a single derivative in each equation. However, we obtain equations having a simple structure in which several derivatives appear in a single equation.