Representation of zero curvature for the system of nonlinear partial differential equations $$x_{\alpha ,z\bar z} = \exp (kx)_\alpha $$ and its integrability

Relativistic Toda systems

“…Meanwhile, distribution of a copy of [12] has given nse to a number of papers containing further information on the ~elati_vistic Toda systems at issue here [13][14][15][16][17][18][19]. (We should add at this point that m ~his paper we have nothing to say about relativistic Toda type field theories, as mtroduced and studied in [20,21]. )…”

C2 TODA THEORY IN THE REDUCED WZNW FRAMEWORK

“…It is known for a long time that this model is soluble. The exact solution of the Toda models has been found by Leznov and Savaliev in [1].…”

“…Analysing the classical Toda models Leznov and Savaliev showed that their equation of motion can be written as zero curvature conditions and in this way they are exactly integrable [1]. Then Bilal and Gervais discovered that the symmetry algebras of the models are the corresponding W algebras [2], extensions of the chiral Virasoro algebra by some higher spin chiral currents [3].…”

Алгебры Симметрий Лагранжевых Систем Лиувиллева Типа

“…Двумер-ные цепочки Тоды u xy = e Ku , связанные с полупростыми комплексными алгебра-ми Ли, -наиболее хорошо изученный пример таких уравнений [1], [2], [5]- [7]. Из-вестно, что системы, принадлежащие рассматриваемому классу, допускают высшие симметрии вида ϕ = (φ), зависящие от свободных функциональных параметров φ = t φ 1 x, [w] , .…”