On the Bäcklund transformation and the theorem of permutability for the generalized Weierstrass system

Bracken, Paul; Grundland, A. M.

doi:10.1088/0266-5611/16/1/312

Cited by 7 publications

(9 citation statements)

References 16 publications

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“…The compatibility condition for (1.9) reproduces the system (1.8) in the variable q. It has also been shown [15], that for a chosen solution q of (1.8), the symmetry group G of the overdetermined system (1.8) and (1.9) with two-dimensional orbits has a complete set of two functionally independent invariants. Thus, the solution p of the initial equations (1.8) and (1.9) can be expressed in terms of these invariants.…”

Section: )mentioning

confidence: 82%

“…It was shown using the conditional symmetry method [15], that the GW system admits an AutoBäcklund transformation for any holomorphic function J, ∂p = −λqp 2 + ∂q q p − λJ q ,∂J = 0, 9) where the function q satisfies the elliptic Sh-Gordon equation (1.8). The arbitrary complex constant λ is the Bäcklund parameter.…”

Section: )mentioning

confidence: 99%

“…The permutability theorem for the Auto-Bäcklund transformation for system (1.3) has recently been formulated by the authors [15] and new classes of nonsplitting multi-soliton solutions of system (1.3) have been obtained. The objective of this paper is a systematic analysis of GW system (1.3) from the symmetry group point of view.…”

Section: )mentioning

confidence: 99%

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Symmetry properties and explicit solutions of the generalized Weierstrass system

Bracken

Grundland

2001

Journal of Mathematical Physics

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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 Painlevé 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 systems to certain differential constraints. In this case, several types of nonsplitting algebraic, trigonometric and hyperbolic multi-soliton solutions have been obtained in explicit form. A new procedure for constructing solutions of the overdetermined system which is composed of the generalized Weierstrass system and the complex eikonal equations is studied. Finally, an approach to the classical configurations of strings in three-dimensional Euclidean space based on the obtained solutions of the generalized Weierstrass system is presented. RésuméLa méthode de réduction par symétries est appliquée systématiquement pour dériver plusieurs classes de solutions invariantes du système de Weierstrass généralisé induisant des surfaces de courbure moyenne constante et du modèle sigma euclidien bidimensionnel associéà ce système. Une classification des sous-groupes avec orbites génériques de codimension un des groupes de Lie de symétrie ponctuels pour ces systèmes fournit un moyen d'introduire des variables de symétrie et réduit le système initialà différents systèmes inéquivalents d'équations différentielles ordinaires. Nous effectuons une analyse des singularités afin d'établir si ceséquations différentielles ordinaires possèdent la propriété de Painlevé. Ceséquations différentielles ordinaires peuvent alorsêtre transformées dans des formes standards et ensuite résolues en termes de fonctionsélémentaires et de fonctions elliptiques de Jacobi. Cela résulte en un grand nombre de nouvelles solutions et, dans certains cas, de nouvelles surfacesà courbure moyenne constante sont trouvées. De plus, cette analyse des symétries estétendue au cas des symétries conditionnelles en soumettant les systèmes initiauxà certaines contraintes différentielles. Dans ce cas, plusieurs solutions multi-solitoniques non-séparantes de type algébrique, trigonomé-trique et elliptique sont obtenues sous une forme explicite. Une nouvelle procédure pour construire des solutions du système surdéterminé composé du ...

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Section: )mentioning

confidence: 82%

Section: )mentioning

confidence: 99%

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Symmetry properties and explicit solutions of the generalized Weierstrass system

Bracken

Grundland

2001

Journal of Mathematical Physics

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“…The most important advantage of this method is that it gives us effective tools for constructing certain classes of BT described by first order differential equations in a systematic way. Its effectiveness has been demonstrated by the results, both reconstructed [10,21,24,25] and new in Section 3 and in Section 4 for the GW system (3.1).…”

Section: Final Remarksmentioning

confidence: 99%

“…Hence from equations (3.1) and (3.3), we can determine the induced metric of this surface [23,24]. However, the matrix appearing in the Lax pair is singular and nilpotent, which prevents the construction of solutions of (3.1).…”

Section: The Generalized Weierstrass Systemmentioning

confidence: 99%

On Complete Integrability of the Generalized Weierstrass System

Bracken

Grundland

2002

JNMP

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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.

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The generalized Weierstrass system and an application to the study of deformations of surfaces by means of integrable hierarchies

Bracken¹

2009

Dynamical Systems

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On the Bäcklund transformation and the theorem of permutability for the generalized Weierstrass system

Cited by 7 publications

References 16 publications

Symmetry properties and explicit solutions of the generalized Weierstrass system

Symmetry properties and explicit solutions of the generalized Weierstrass system

On Complete Integrability of the Generalized Weierstrass System

The generalized Weierstrass system and an application to the study of deformations of surfaces by means of integrable hierarchies

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