Aesthetic Field Equations Research Articles

Irregular oscillations and mathematical aesthetics

We have shown that a set of mathematically aesthetic principles can be written down and that these principles can be cast into a set of partial differential equations. The question is what sort of information is contained in these equations. Previously, we showed that for a certain choice of origin point data the equations collapse into a simple system that describes sinusoidal behavior along any integration path segment. For another set of origin point data, the equations collapse into a system that describes a sine within sine curve along any path segment. Here we show that for a more general set of origin point data the equations describe irregular oscillations along the coordinate axes for both the small picture and the big picture. From this we infer that the aesthetic field equations describe irregular oscillations along any path segment (at least close to the origin). The solution under study is a wave packet solution. The wave packet solution is characterized by regions of large magnitude immersed in a “vacuum” that appears empty but, in fact, has considerable structure. The wave packet solution is studied when an integration path is specified.

Oscillations within oscillations

Previously we have shown that for a particular choice of origin point data, the nonintegrable Aesthetic Field Equations collapse into a simpler set of equations, called the A, B, J, L equations. These equations lead to sinusoidal variation along any path segment for all the field quantities. In this paper we find, for a different choice of origin point data, that the Aesthetic Field Equations simply into a nonlinear system that describes a sine curve within a sine curve along any path segment. These results are obtained from visual inspection of the computer plots as well as numerical fitting to the data. We examine two dimensional maps, first when we specify an integration path, and then when we make use of a superposition principle at each point. The superposition principle arises as a consequence of the theory of nonintegrable systems, which we have developed on an earlier occasion. We find two dimensional maps in both instances that do not appear regular in the small. However, in both cases, we were able to observe large scale regularities. In our studies, we make use of computer techniques that focus on the large wavelength oscillations (big picture) as well as techniques that focus on the small wavelength oscillations (small picture). When the superposition principle is used, the small as well as big oscillations no longer have a sinusoidal appearance for y equals constant lines. This contrasts with the case when we specify a path where we see a sine curve within a sine curve along these y equals constant lines. In addition, we obtain a set of linear equations that describes a sine curve within a sine curve that can be considered in its own right. In this case, in addition to numerical integrations, we show analytically that a sine curve within a sine curve is a solution to the equations.

Computer two dimensional maps of loop soliton lattice systems using the new approach to the no integrability Aesthetic field equations

We show that there are varieties of somewhat different loop soliton lattices when we specify an integration path in No Integrability Aesthetic Field Theory. These are illustrated using two dimensional computer maps. We have previously studied several such systems using the new approach to non‐integrable systems developed in previous papers. [1‐3]. The results of these earlier papers indicated that the solitons were rearranged by the new integration scheme in an erratic looking manner. However, we were restricted to regions close to the origin in these studies. With additional computer time made available and the use of tapes to store large amounts of information we have studied the above loop soliton systems and have been able to map considerable larger regions of the x, y plane. Symmetries for locations of the planar maxima and minima have been uncovered within a particular quadrant, although the symmetry found is not as great as the lattice. The type symmetry found is not maintained when more than one quadrant is involved. We also study a system that can be looked at as a perturbation of a loop lattice system. A brief discussion of the background material appears in the appendix.

Summation-over-paths degree of freedom and the loop lattice

Previously we found different kinds of lattice solutions in the no-integrability aesthetic field theory when we specify an integration path. When we combine the lattice origin-point data and aesthetic field equations with the summation-over-paths degree of freedom, the lattice solution is altered in a dramatic way. In this paper we study particular how a loop lattice system is affected by the summation over paths. We find likely evidence for two loop particles. The other potential particle systems could be closed strings (loops), but increased computer capability would be necessary to confirm this.

Study of different lattice solutions in aesthetic field theory

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.

Aesthetic fields: a two-dimensional lattice solution satisfying integrability

In our previous paper we obtained examples of lattice solutions to the aesthetic field equations. A drawback with these solutions was that the integrability equations were not satisfied. In this paper we find a solution to the aesthetic field equations which describes a two-dimensional lattice structure. The integrability equations are satisfied in this case. We work within a six-dimensional framework in this paper. Three of the coordinates are real and three of the coordinates are pure imaginary. It is a generalization of the Minkowski version of aesthetic field theory.

Aesthetic fields: A topological particle

We extend the complex aesthetic field equations forAjki andBjk/i so as to have symmetry between these fields. We find computer solutions in which the confluence region takes on a loop shape. We call such a particle a “topological” particle.

Sinusoidal solutions to the aesthetic field equations

The aesthetic field equations do not resemble the wave equation, nor was the motivation behind them the wave equation. Nevertheless, we show that there exists a solution to the field equations that satisfies the wave equation. Integrability is also satisfied by this solution. Previously we showed that the Aesthetic Field Equations have particle solutions. Now we see that the equations also have sinusoidal solutions.

A study of solutions to the aesthetic field equations

We have found hundreds of solutions to the integrability equations in aesthetic field theory. The behavior of the solutions to the aesthetic field equations depends on which solution to the integrability equations we take. From computer runs down a coordinate axis we have found a type of solution where we have a maximum and a minimum, as well as the field going to zero at large distances along both directions. This kind of solution is quite prevalent. We call this type of solution a “pulse” solution. We have found the “pulse” solution in two and three dimensions as well as four dimensions. It appears regardless of whether certain symmetries are present or absent. We have taken a two- or three-dimensional Гαβγ and made a four-dimensional theory from it with the use of a four-dimensionaleαi. This process we call “imbedding.” We have found imbedding has not affected the overall characteristics of the solution in the cases we considered. We were able to change the character of the solutions to some degree by altering the magnitude of some of the gammas—but this did not lead to solutions with significantly more wiggles. We also found an example of an oscillatory solution. The oscillations occurred in too regular a pattern to give a realistic model for basic behavior. However, this solution indicates that aesthetic field theory has more structure then we have ever seen before. We also obtained a solution in which errors took over so fast that the computer was literally helpless in telling us what is going on. In other solutions the field appears to increase without bounds. Whether this is due to singularities or to the presence of large numbers is not clear.

A new particle solution in aesthetic field theory

In present theories a particle is commonly associated with a singularity of the field. A more realistic picture would describe the particle by an intense but singularity-free field. We have found a new solution to the aesthetic field equations for which the field associated with the particle has a very large magnitude. The particle appears to be bounded despite the large numbers appearing in the solution. We prove that this present solution is not equivalent to theO(3)-invariant solution discussed in Muraskin (1973b). Since our present solution appears well-behaved, the suggestion is that we do not confine ourselves toO(3)-invariant data in future work. Owing to the large magnitude fields, we were unable to study trajectories of the particle in any detail. There is nothing wrong, in itself, with large numbers. The present solution, which we have now studied, is the first instance in our work on aesthetic field theory in which large numbers appear without the suggestion of unboundedness.

New solutions in aesthetic field theory

We have obtained a large number of solutions to the aesthetic field equations. We discuss 19 solutions which appear to lead to bounded particle systems. One of the solutions is more complex (although only slightly) than the solution discussed in detail in Muraskin and Ring (1975). The solutions we have found have varied mathematical properties.

A two-particle collision in aesthetic field theory

We have found a new computer solution to the aesthetic field equations. This solution describes a two-particle system with more structure than previously found. The contour lines show an arm structure. We have observed four arms around the maximum center. The location of the maximum (minimum) center is not along a straight line as a function of time. This is the first time that such an effect has been observed for any kind of nonlinear partial differential equation, so far as we know. A further discussion of the aesthetic principles leading to the field equations is given.