Soliton surfaces associated with symmetries of ODEs written in Lax representation

The main aim of this paper is to discuss recent results on the adaptation of the Fokas-Gel'fand procedure for constructing soliton surfaces in Lie algebras, which was originally derived for PDEs [Grundland, Post 2011], to the case of integrable ODEs admitting Lax representations. We give explicit forms of the \g-valued immersion functions based on conformal symmetries involving the spectral parameter, a gauge transformation of the wave function and generalized symmetries of the linear spectral problem. The procedure is applied to a symmetry reduction of the static $\phi^4$-field equations leading to the Jacobian elliptic equation. As examples, we obtain diverse types of surfaces for different choices of Jacobian elliptic functions for a range of values of parameters.Comment: 14 Pages, 2 figures Conference Proceedings for QST7 Pragu

Properties of soliton surfaces associated with integrable CPN−1\mathbb{C}P^{N-1} sigma models

We investigate certain properties of $\mathfrak{su}(N)$-valued two-dimensional soliton surfaces associated with the integrable $\mathbb{C}P^{N-1}$ sigma models constructed by the orthogonal rank-one Hermitian projectors, which are defined on the two-dimensional Riemann sphere with finite action functional. Several new properties of the projectors mapping onto one-dimensional subspaces as well as their relations with three mutually different immersion formulas, namely, the generalized Weierstrass, Sym-Tafel and Fokas-Gel'fand have been discussed in detail. Explicit connections among these three surfaces are also established by purely analytical descriptions and, it is demonstrated that the three immersion formulas actually correspond to the single surface parametrized by some specific conditions.Comment: 17 page

On analytic descriptions of two-dimensional surfaces associated with the CP^(N-1) sigma model

We study analytic descriptions of conformal immersions of the Riemann sphere S^2 into the CP^(N-1) sigma model. In particular, an explicit expression for two-dimensional (2-D) surfaces, obtained from the generalized Weierstrass formula, is given. It is also demonstrated that these surfaces coincide with the ones obtained from the Sym-Tafel formula. These two approaches correspond to parametrizations of one and the same surface in R^(N^2-1).Comment: 6 page