arXiv subjects

P. Bracken

Publications and source records attributed to P. Bracken.

3 recordsLinked to original sources

arXiv2003

Solutions of the generalized Weierstrass representation in four-dimensional Euclidean space

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.

nlin.SI↗

arXiv2002

On complete integrability of the generalized Weierstrass system

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 $2 \times 2$ matrices based on an $sl (2, \mathbb C)$ 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.

nlin.SI↗

arXiv2001

Multivortex Solutions of the Weierstrass Representation

The connection between the complex Sine and Sinh-Gordon equations on the complex plane associated with a Weierstrass type system and the possibility of construction of several classes of multivortex solutions is discussed in detail. We perform the Painlevé test and analyse the possibility of deriving the Bäcklund transformation from the singularity analysis of the complex Sine-Gordon equation. We make use of the analysis using the known relations for the Painlevé equations to construct explicit formulae in terms of the Umemura polynomials which are $τ$-functions for rational solutions of the third Painlevé equation. New classes of multivortex solutions of a Weierstrass system are obtained through the use of this proposed procedure. Some physical applications are mentioned in the area of the vortex Higgs model when the complex Sine-Gordon equation is reduced to coupled Riccati equations.

nlin.SI↗