Symmetry properties and explicit solutions of the generalized Weierstrass system

Abstract

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 ...

SusyCPN−1model and surfaces in

“…This has generated interest 3,4 in looking at the properties of these surfaces and relating them to the solutions This approach involved writing the equation for the harmonic map as a conservation law and then observing that in this construction a special operator played a key role. This operator, related to the fundamental projector of the harmonic map was then used in the construction of the surface.…”

On CP 1 and CP 2 Maps and Weierstrass Representations for Surfaces Immersed into Multi-Dimensional Euclidean Spaces

“…The converse is also true [21]. Thus, if w is a solution of (3.5), then ψ 1 and ψ 2 of the GW system (3.1) have the form…”

“…For the sake of convenience our investigation starts with a derivation of the position vector X(z,z) of a surface in R 3 from the Lax pair for a GW system (3.1). As it was shown in [21] the GW system (3.1) is in a one-to-one correspondence with the solutions of the equations of the completely integrable two-dimensional Euclidean CP 1 sigma model…”

Intrinsic Formulation of Geometric Integrability and Associated Riccati System Generating Conservation Laws

“…This is the case for both constant mean curvature surfaces as well as surfaces which have constant Gaussian curvature [2]. It has been shown that constant mean curvature surfaces play an important role in soliton theory by means of the generalized Weierstrass representation of Konopelchenko [2]. Moreover, Sasaki [3,4] established a geometrical interpretation for the inverse-scattering problem which was originally formulated by Ablowitz [5] and associates in terms of pseudospherical surfaces [6].…”

“…Such equations have Bäcklund transformations [1] and moreover their solutions can be associated with the generation of such geometrical objects as surfaces. This is the case for both constant mean curvature surfaces as well as surfaces which have constant Gaussian curvature [2]. It has been shown that constant mean curvature surfaces play an important role in soliton theory by means of the generalized Weierstrass representation of Konopelchenko [2].…”