A formula for constructing infinitely many surfaces on Lie algebras and integrable equations

Abstract

Surfaces immersed in Lie algebras can be characterized by the so called fundamental forms. The coefficients of these forms satisfy a system of nonlinear partial differential equations (PDEs), the Gauss-Mainardi-Codazzi-Ricci equations. For particular surfaces, this system of PDEs belongs to a distinguished class of equations known as integrable equations. Such an example in R 3 is the class of surfaces of constant mean curvature which is associated with the sinh-Gordon equation. Here an explicit formula is presented which associates with a given system of integrable nonlinear PDEs infinitely many surfaces immersed in Lie algebras. This formula is based on the general construction of surfaces on Lie algebras introduced recently by the first two authors, and on the fact that integrable equations possess infinitely many symmetries. Several examples of surfaces immersed in the 3-dimensional Euclidean space are discussed, including the list of integrable surfaces recently presented by Bobenko and certain deformations thereof. Mathematics Subject Classification (2000). 35Q53, 35Q58, 53A05.

On Immersion Formulas for Soliton Surfaces

“…In reference [9], the authors looked for a simultaneous infinitesimal deformation of the associated LSP (2.15) which preserves the ZCC (2.14) 19) where [ , ] is the Lie algebra commutator and c k ij are the structural constants of g. Since this generator X e does not transform λ, these symmetries preserve the singularity structure of the potential matrices U α in the spectral parameter λ. The infinitesimal deformation of the LSP (2.13) and the ZCC (2.14) under the infinitesimal transformation (2.16) requires that the matrix functions U α and Ψ satisfy, at first order in , the equations…”

“…All these symmetries can be used to determine explicitly a g-valued immersion function F of a 2D-surface. Thus, a generalization of the FG formula for immersion can be formulated as follows [9,11].…”

“…Another approach for finding such surfaces has been formulated by J. Cieslinski and A. Doliwa [5][6][7] using gauge symmetries of the LSP. A third approach, using the LSP for integrable systems and their symmetries has been introduced by Fokas and Gel'fand [8,9], to construct families of soliton surfaces. Most recently, in a series of papers [11][12][13]15], a reformulation and extension of the Fokas-Gel'fand immersion formula has been performed through the formalism of generalized vector fields and their actions on jet spaces.…”

Surfaces and curves corresponding to the solutions generated from periodic “seed” of NLS equation

“…Stimulated by many interesting works on the integrable curves and surfaces [2][3][4][5][6][7][8][9][10], we shall in this paper discuss the curves and surfaces corresponding to solutions generated from periodic "seed" q = ce i(as+bt) by T n for the NLS. The main tools are determinant representation of T n and the Sym formula [2] as we have used in [1].…”

Surfaces immersed in Lie algebras obtained from the sigma models

“…For a classical presentation we refer to a treatise by Eisenhardt [9] and for a modern approach to the subject see e.g. [1,2,12,13,26,34,36,37,38] and references therein, in particular the recent books by F. Helein [23,24] and K. Kenmotsu [27]. This topic has been further explored by, among others, B. Konopelchenko [30].…”