Abstract:
Some aspects of anyon physics are reviewed with the intention of establishing a model for the quantization of the Hall conductance. A single particle Schrödinger model is introduced and coupled with a constraint equation formulated from the anyon picture. The Schrödinger equation-constraint system can be converted to a single nonlinear differential equation and solutions for the model can be produced.

Abstract:
Properties of eigenvalues of the p-Laplacian operator on a finite dimensional compact Riemannian manifold are studied for the case in which the metric of the manifold evolves under the Ricci-harmonic map flow. It will be shown that the first nonzero eigenvalue is monotonically nondecreasing along the flow and differentiable almost everywhere.

Abstract:
An important question that arises is which surfaces in three-space admit a mean curvature preserving isometry which is not an isometry of the whole space. This leads to a class of surface known as a Bonnet surface in which the number of noncongruent immersions is two or infinity. The intention here is to present a proof of a theorem using an approach which is based on differential forms and moving frames and states that helicoidal surfaces necessarily fall into the class of Bonnet surfaces. Some other results are developed in the same manner.

Abstract:
The intrinsic geometry of surfaces and Riemannian spaces will be investigated. It is shown that many nonlinear partial differential equations with physical applications and soliton solutions can be determined from the components of the relevant metric for the space. The manifolds of interest are surfaces and higher-dimensional Riemannian spaces. Methods for specifying integrable evolutions of surfaces by means of these equations will also be presented.

Abstract:
The moving frame and associated Gauss-Codazzi equations forsurfaces in three-space are introduced. A quaternionicrepresentation is used to identify the Gauss-Weingarten equationwith a particular Lax representation. Several examples are given,such as the case of constant mean curvature.

Abstract:
The relationship between solutions of the sinh-Laplace equation and the determination of various kinds of surfaces of constant Gaussian curvature, both positive and negative, will be investigated here. It is shown that when the metric is given in a particular set of coordinates, the Gaussian curvature is related to the sinh-Laplace equation in a direct way. The fundamental equations of surface theory are found to yield a type of geometrically based Lax pair for the system. Given a particular solution of the sinh-Laplace equation, this Lax can be integrated to determine the three fundamental vectors related to the surface. These are also used to determine the coordinate vector of the surface. Some specific examples of this procedure will be given.

Abstract:
The symmetry group method is applied to a generalized Korteweg-de Vries equation and several classes of group invariant solutions for it are obtained by means of this technique. Polynomial, trigonometric, and elliptic function solutions can be calculated. It is shown that this generalized equation can be reduced to a first-order equation under a particular second-order differential constraint which resembles a Schrödinger equation. For a particular instance in which the constraint is satisfied, the generalized equation is reduced to a quadrature. A condition which ensures that the reciprocal of a solution is also a solution is given, and a first integral to this constraint is found.

Abstract:
An alternate generalized Korteweg-de Vries system is studied here. A procedure for generating solutions is given. A theorem is presented, which is subsequently applied to this equation to obtain a type of Bäcklund transformation for several specific cases of the power of the derivative term appearing in the equation. In the process, several interesting, new, ordinary, differential equations are generated and studied.

Abstract:
A type of prolongation structure for several general systems is discussed. They are based on a set of one forms for which the underlying structure group of the integrability condition corresponds to the Lie algebra of , , and . Each will be considered in turn and the latter two systems represent larger cases. This geometric approach is applied to all of the three of these systems to obtain prolongation structures explicitly. In both cases, the prolongation structure is reduced to the situation of three smaller problems. 1. Introduction Geometric approaches have been found useful in producing a great variety of results for nonlinear partial differential equations [1]. A specific geometric approach discussed here has been found to produce a very elegant, coherent, and unified understanding of many ideas in nonlinear physics by means of fundamental differential geometric concepts. In fact, relationships between a geometric interpretation of soliton equations, prolongation structure, Lax pairs, and conservation laws can be clearly realized and made use of. The interest in the approach, its generality, and the results it produces do not depend on a specific equation at the outset. The formalism in terms of differential forms [2] can encompass large classes of nonlinear partial differential equation, certainly the AKNS systems [3, 4], and it allows the production of generic expressions for infinite numbers of conservation laws. Moreover, it leads to the consequence that many seemingly different equations turn out to be related by a gauge transformation. Here the discussion begins by studying prolongation structures for a system discussed first by Sasaki [5, 6] and Crampin [7] to present and illustrate the method. This will also demonstrate the procedure and the kind of prolongation results that emerge. It also provides a basis from which to work out larger systems since they can generally be reduced to problems. Of greater complexity are a pair of problems which will be considered next. It is shown how to construct an system based on three constituent one forms as well as an system composed of eight fundamental one forms. The former has not appeared. The problem is less well known than the problem; however, some work has appeared in [8, 9]. In the first of these, the solitons degenerate to the AKNS solitons in three ways, while in the latter nondegenerate case, they do not. All of the results are presented explicitly; that is, the coefficients of all the forms and their higher exterior derivatives are calculated and given explicitly. Maple is used to do this