Exact Results in Quantum Theory

In this talk, we investigate certain features of generalized symmetries of integrable systems in order to construct the Fokas-Gel'fand formula for the immersion of 2D-soliton surfaces in Lie algebras. We demonstrate that if there exists a common symmetry of the zero-curvature representation of an integrable PDE and its linear spectral problem then the Fokas-Gel'fand immersion formula is applicable in its original form. In the general case, we show that when a symmetry of the zero-curvature representation is not necessarily a symmetry of its linear spectral problem, the immersion function of a 2D-surface is governed by an extended formula involving additional terms in the expression for the tangent vectors. We show that the Sym-Tafel formula for the immersion of soliton surfaces in Lie algebras can be mapped to its counterparts, the Cieslinski-Doliwa formula and the Fokas-Gelfand formula. Finally, we illustrate these results by examples involving an elliptic ODE and the CP^N-1 sigma model equation.