Paul Bracken

The Weierstrass–Enneper system for constant mean curvature surfaces and the completely integrable sigma model

The integrability of a system which describes constant mean curvature surfaces by means of the adapted Weierstrass–Enneper inducing formula is studied. This is carried out by using a specific transformation which reduces the initial system to the completely integrable two-dimensional Euclidean nonlinear sigma model. Through the use of the apparatus of differential forms and Cartan theory of systems in involution, it is demonstrated that the general analytic solutions of both systems possess the same degree of freedom. Furthermore, a new linear spectral problem equivalent to the initial Weierstrass–Enneper system is derived via the method of differential constraints. A new procedure for constructing solutions to this system is proposed and illustrated by several elementary examples, including a multi-soliton solution.

Symmetry properties and explicit solutions of the generalized Weierstrass system

The method of symmetry reduction is systematically applied to derive several classes of invariant solutions for the generalized Weierstrass system inducing constant mean curvature surfaces and to the associated two-dimensional nonlinear sigma model. A classification of subgroups with generic orbits of codimension one of the Lie point symmetry group for these systems provides a tool for introducing symmetry variables and reduces the initial systems to different nonequivalent systems of ordinary differential equations. We perform a singularity analysis for them in order to establish whether these ordinary differential equations have the Painleve property. These ordinary differential equations can then be transformed to standard forms and next solved in terms of elementary and Jacobi elliptic functions. This results in a large number of new solutions and in some cases new interesting constant mean curvature surfaces are found. Furthermore, this symmetry analysis is extended to include conditional symmetries by subjecting the original system to certain differential constraints. In this case, several new types of nonsplitting algebraic, trigonometric, and hyperbolic multisoliton solutions have been obtained in explicit form. Some physical interpretation of these results in the areas of fluid membranes, string theory, two-dimensionl gravity, and cosmology are given.

An arithmetic-geometric mean inequality

Abstract Several integrals which are related to the arithmetic-geometric mean are developed and proved in a very elementary way. These results can be used to prove a known inequality which relates this mean to the logarithmic mean.

Symmetry properties of a generalized Korteweg-de Vries equation and some explicit solutions

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 Schrodinger 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.

On Certain Classes of Solutions of the Weierstrass-Enneper System Inducing Constant Mean Curvature Surfaces

Abstract Analysis of the generalized Weierstrass-Enneper system includes the estimation of the degree of indeterminancy of the general analytic solution and the discussion of the boundary value problem. Several different procedures for constructing certain classes of solutions to this system, including potential, harmonic and separable types of solutions, are proposed. A technique for reduction of the Weierstrass-Enneper system to decoupled linear equations, by subjecting it to certain differential constraints, is presented as well. New elementary and doubly periodic solutions are found, among them kinks, bumps and multi-soliton solutions.

Relativistic Equations of Motion from Poisson Brackets

The inverse problem of Poisson dynamics isreviewed as well as a derivation of the Maxwellequations from a postulated set of Poisson brackets. Theformalism is extended to the relativistic case bypostulating Poisson brackets, as in the nonrelativisticcase, and using the relativistic Hamiltonian. A systemof relativistic equations of motion is obtained, and itis indicated that a system of consistency conditions remains valid in this limit.

Hamiltonians for the Quantum Hall Effect on Spaces with Non-Constant Metrics

The problem of studying the quantum Hall effect on manifolds with non constant metric is addressed. The Hamiltonian on a space with hyperbolic metric is determined, and the spectrum and eigenfunctions are calculated in closed form. The hyperbolic disk is also considered and some other applications of this approach are discussed as well.

A geometric interpretation of prolongation by means of connections

A geometric interpretation of prolongation can be formulated by using the theory of connections. A fiber bundle can be established which is composed of a base manifold and variables which span a prolongation space. A particular connection is introduced in terms of these coordinates. This provides a very different way of viewing the technique and for introducing prolongation algebras as well as generating integrable equations in a novel way.

Solutions of the Generalized Weierstrass Representation in Four-Dimensional Euclidean Space

Abstract Several classes of solutions of the generalized Weierstrass system, which induces constant mean curvature surfaces into four-dimensional Euclidean space are constructed. A gauge transformation allows us to simplify the system considered and derive factorized classes of solutions. A reduction of the generalized Weierstrass system to decoupled CP 1 sigma models is also considered. A new procedure for constructing certain classes of solutions, including elementary solutions (kinks and bumps) and multisoliton solutions is described in detail. The constant mean curvature surfaces associated with different types of solutions are presented. Some physical interpretations of the results obtained in the area of string theory are given.

On Complete Integrability of the Generalized Weierstrass System

Abstract In this paper we study certain aspects of the complete integrability of the Generalized Weierstrass system in the context of the Sinh-Gordon type equation. Using the conditional symmetry approach, we construct the Bäcklund transformation for the Generalized Weierstrass system which is determined by coupled Riccati equations. Next a linear spectral problem is found which is determined by nonsingular 2×2 matrices based on an sl(2, ℂ) representation. We derive the explicit form of the Darboux transformation for the Weierstrass system. New classes of multisoliton solutions of the Generalized Weierstrass system are obtained through the use of the Bäcklund transformation and some physical applications of these results in the area of classical string theory are presented.