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Paul Bracken

Publications and source records attributed to Paul Bracken.

18 recordsLinked to original sources
arXiv2014

Quantization of a Particle on a Two-Dimensional Manifold of Constant Curvature

The formulation of quantum mechanics on spaces of constant curvature is studied. It is shown how a transition from a classical system to the quantum case can be accomplished by the quantization of the Noether momenta. These can be determined by Lie differentiation of the metric which defines the manifold. For the metric examined here, it is found that the resulting Schrodinger equation is separable and the spectrum and eigenfunctions can be investigated in detail.

math-ph↗
arXiv2014

Motion on Constant Curvature Spaces and Quantization Using Noether Symmetries

A general approach is presented for quantizing a metric nonlinear system on a manifold of constant curvature. It makes use of a curvature dependent procedure which relies on determining Noether symmetries from the metric. The curvature of the space functions as a constant parameter. For a specific metric which defines the manifold, Lie differentiation of the metric gives these symmetries. A metric is used such that the resulting Schrodinger equation can be solved in terms of hypergeometric functions. This permits the investigation of both the energy spectrum and wave functions exactly for this system.

math-ph↗
arXiv2014

Connections of Zero Curvature and Applications to Nonlinear Partial Differential Equations

A general formulation of zero curvature connections in a principle bundle is presented and some applications are discussed. It is proved that a related connection based on a prolongation in an associated bundle remains zero curvature as well. It is also shown that the connection coefficients can be defined so that the partial differential equation to be studied appears as the curvature term in the structure equations. It is discussed how Lax pairs and Backlund transformations can be formulated for such equations. It is discussed how Lax pairs and Backlund transformations can be formulated for such equations that occur as zero curvature terms.

math.DG↗
arXiv2013

An Intrinsic Characterization of Bonnet Surfaces Based on a Closed Differential Ideal

The structure equations for a surface are introduced and two required results based on the Codazzi equations are obtained from them. Important theorems pertaining to isometric surfaces are stated and a theorem of Bonnet is obtained. A tranformation formula for the connection forms is developed. It is proved that the angle of deformation must be harmonic. It is shown that the differentials of many of the important variables generate a closed differential ideal. This implies that a coordinate system exists in which many of the variables satisfy particular ordinary differential equations and these results can be used to characterize Bonnet surfaces.

math.DG↗
arXiv2011

Geometric Approaches for Generating Prolongations for Nonlinear Partial Differential Equations

The prolongation structure of a two-by-two problem is formulated very generally in terms of exterior differential forms on a standard representation of Pauli matrices. The differential system is general without making reference to any specific equation. An integrability condition is provided which gives by construction the equation to be investigated and whose components involve the structure constants of an SU(2) Lie algebra. Along side this, a related, different kind of prolongation, a type of Wahlquist-Estabrook prolongation, over a closed differential ideal is discussed and some applications are given.

math-ph↗
arXiv2011

A Geometric Method to Investigate Prolongation Structures for Differential Systems With Applications to Integrable Systems

A type of prolongation structure for several general systems is discussed. They are based on a set of one-forms in which the underlying structure group of the integrability condition corresponds to the Lie-algebra of SL (2,R), O(3), or SU(3). Each will be considered in turn and the latter two systems represent larger 3by3 cases. This geometric approach is applied to all three of these systems to obatin prolongation structures explicitly. In both 3by3 cases the prolongation structure is reduced to the situation of three smaller 2by2 problems. Many types of conservation laws can be obtained at different stages of the development, and at the end, a single result is developed to show how this can be done.

math-ph↗
arXiv2011

The Einstein-Hilbert Action Horizons and Connections with Thermodynamics

It is shown that the Einstein-Hilbert action can be constructed by minimizing free energy. The entropy used to determine the free energy is determined on the horizon of a black hole. Some further considerations with regard to generalizations of these ideas to other situations of physical importance are presented as well.

gr-qc↗
arXiv2009

Integrable Systems of Partial Differential Equations Determined by Structure Equations and Lax pair

It is shown how a system of evolution equations can be developed both from the structure equations of a submanifold embedded in three-space as well as from a matrix SO(6) Lax pair. The two systems obtained this way correspond exactly when a constraint equation is selected and imposed on the system of equations. This allows the coefficients of the second fundamental form to be selected in a more general way so they need not be constants.

math-ph↗
arXiv2009

The interrelationship of integrable equations, differential geometry and the geometry of their associated surfaces

A survey of some recent and important results which have to do with integrable equations and their relationship with the theory of surfaces is given. Some new results are also presented. The concept of the moving frame is examined, and it is used in several subjects, which are discussed. Structure equations are introduced in terms of differential forms. Forms are shown to be very useful in relating geometry, equations and surfaces, which appear in many sections. The topics of the chapters are different and separate, but joined together by common themes and ideas. Several subjects which are not easy to access are elaborated, such as Maurer-Cartan cocycles and recent results with regard to generalizations of the Weierstrass-Enneper method for generating constant mean curvature surfaces in three and higher dimensional Euclidean spaces.

math-ph↗
arXiv2009

Exterior Differential Systems, Prolongations and the Integrability of Two Nonlinear Partial Differential Equations

A generalized KdV equation is formulated as an exterior differential system, which is used to determine the prolongation structure of the equation. The prolongation structure is obtained for several cases of the variable powers, and nontrivial algebras are determined. The analysis is extended to a differential system which gives the Camassa-Holm equation as a particular case. The subject of conservation laws is briefly discussed for each of the equations. A Backlund transformation is determined using one of the prolongations.

math-ph↗
arXiv2007

A Chiral Schwinger model, its Constraint Structure and Applications to its Quantization

The Jackiw-Rajaraman version of the chiral Schwinger model is studied as a function of the renormalization parameter. The constraints are obtained and they are used to carry out canonical quantization of the model by means of Dirac brackets. By introducing an additional scalar field, it is shown that the model can be made gauge invariant. The gauge invariant model is quantized by establishing a pair of gauge fixing constraints in order that the method of Dirac can be used.

math-ph↗
arXiv2006

Dynamics of Induced Surfaces in Four-Dimensional Euclidean Space

The Davey Stewartson hierarchy will be developed based on a set of three matrix differential operators. These equations will act as evolution equations for different types of surface deformation in Euclidean four space. The Weierstrass representation for surfaces will be developed and its uniqueness up to gauge transformations will be reviewed. Applications of the hierarchy will be given with regard to generating deformations of surfaces, and it will be shown that the Willmore functional is preserved under this kind of deformation.

math-ph↗
arXiv2006

A Time Dependent Version of the Quantum WKB Approximation

The phenomenon of quantum tunneling is reviewed and an overview of applying approximate methods for studying this effect is given. An approach to a time-dependent formalism is proposed in one dimension and generalized to higher dimensions. Some physical examples involving the resulting wavefunction which is determined are presented.

math-ph↗
arXiv2006

Symmetry Properties of a Generalized Korteweg-de Vires Equation and some Explicit Solutions

The symmetry group method is applied to a generalized Korteweg-de Vries equation and several classes of group invarint solution 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.

math-ph↗
arXiv2006

Determination of the Electromagnetic Lagrangian from a System of Poisson Brackets

The Lagrangian and Hamiltonian formulations of electromagnetism are reviewed and the Maxwell equations are obtained from the Hamiltonian for a system of many electric charges. It is shown that three of the equations which were obtained from the Hamiltonian, namely the Lorentz force law and two Maxwell equations, can be obtained as well from a set of postulated Poisson brackets. It is shown how the results derived from these brackets can be used to reconstruct the original Lagrangian for the theory aided by some reasoning based on physical concepts.

math-ph↗
arXiv1999

On certain classes of solutions of the Weierstrass-Enneper system inducing constant mean curvature surfaces

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.

math.AP↗