Induced Surfaces and Their Integrable Dynamics Ii. Generalized Weierstrass Representations in 4‐d Spaces and Deformations via Ds Hierarchy

Abstract. In this paper, we discuss some specific features of symmetries of integrable systems which can be used to contruct the Fokas-Gel'fand formula for the immersion of 2D-soliton surfaces, associated with such systems, in Lie algebras. We establish a sufficient condition for the applicability of this formula. This condition requires the existence of two vector fields which generate a common symmetry of the initial system and its corresponding linear spectral problem. This means that these two fields have to be group-related and we determine an explicit form of this relation. It provides a criterion for the selection of symmetries suitable for use in the Fokas-Gel'fand formula. We include some examples illustrating its application.

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“…Substituting (25) into (21) and using (18) we obtain the tangent vectors D α F in the form postulated by Theorem 1…”

Section: The Sufficient Conditionsmentioning

confidence: 99%

Soliton surfaces and generalized symmetries of integrable systems

Grundland¹,

Post²,

Riglioni³

2013

J. Phys. A: Math. Theor.

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“…It is also invariant under the conformal transformations 10) where α, β : R → R are arbitrary 1-to-1 maps such that ∂ L α(ξ L ) = 0, ∂ R β(ξ R ) = 0, as well as under the parity transformation…”

Section: Cp N −1 Sigma Models and Their Euler-lagrange Equationsmentioning

confidence: 99%

Description of surfaces associated with CPN−1 sigma models on Minkowski space

Grundland

Šnobl

2006

Journal of Geometry and Physics

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The objective of this paper is to construct and investigate smooth orientable surfaces in R N 2 −1 by analytical methods. The structural equations of surfaces in connection with CP N −1 sigma models on Minkowski space are studied in detail. This is carried out using moving frames adapted to surfaces immersed in the su(N ) algebra. The first and second fundamental forms of this surface as well as the relations between them as expressed in the Gauss-Weingarten and Gauss-Codazzi-Ricci equations are found. The Gaussian curvature, the mean curvature vector and the Willmore functional expressed in terms of a solution of CP N −1 sigma model are obtained. An example of a surface associated with the CP 1 model is included as an illustration of the theoretical results.

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“…Recently, such generalization of the GW system has been achieved by B. Konopelchenko et al [12,[43][44][45] where, in particular, the explicit representations for generic surfaces conformally immersed into multi-dimensional Euclidean and pseudo-Euclidean spaces with different signatures have been derived.…”

Section: Application To Classical String Theorymentioning

confidence: 99%

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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Induced Surfaces and Their Integrable Dynamics Ii. Generalized Weierstrass Representations in 4‐d Spaces and Deformations via Ds Hierarchy

Cited by 43 publications

References 41 publications

Soliton surfaces and generalized symmetries of integrable systems

Soliton surfaces and generalized symmetries of integrable systems

Description of surfaces associated with CPN−1 sigma models on Minkowski space

Symmetry properties and explicit solutions of the generalized Weierstrass system

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