On Some Class of Multidimensional Nonlinear Integrable Systems A

On the base of Lie algebraic and differential geometry methods, a wide class of multidimensional nonlinear integrable systems is obtained, and the integration scheme for such equations is proposed. 1. In the report we give a Lie algebraic and differential geometry derivation of a wide class of nonlinear integrable systems of partial differential equations for the functions depending on an arbitrary number of variables, and construct, following the lines of Refs. 1–3, their general solutions in a 'holomorphically factorisable' form. The systems are generated by flat connections, constrained by the relevant grading condition, with values in an arbitrary reductive complex Lie algebra G endowed with a Z–gradation. They describe a multidimen-sional version of Toda type fields coupled to matter fields, and, analogously to the two dimensional situation, with an appropriate Inönü–Wigner contraction procedure, for our systems one can exclude back reaction of the matter fields on the Toda fields. For two dimensional case and the connection taking values in the local part of a finite dimensional algebra G, our equations describe an (abelian and nonabelian) conformal Toda system and its affine deformations for an affine G, see Ref. 1 and references therein, and also Ref. 2 for differential and algebraic geometry background of such systems. For the connection with values in higher grading subspaces of G one deals with systems discussed in Refs. 3, 4. In higher dimensions our systems, under some additional specialisations, contain as particular cases the Cecotti–Vafa type equations 5 written there for a case of the complexified orthogonal algebra, see also Ref. 6; and those of Gervais– Matsuo 7 which represent some reduction of a generalised WZNW model. Due to the lack of space we present here only an announcement of the results which will be described in detail, together with some remarkable examples elsewhere.