Mikhail V. Saveliev

On a solitonic specialisation for the general solutions of some two-dimensional completely integrable systems

Abstract Following a prescription of Olive, Turok and Underwood for a solitonic specialisation of the general solutions to the (abelian) periodic Toda field theories, we discuss a construction of the soliton solutions for a wide class of two-dimensional completely integrable systems arising in the framework of the group-algebraic approach, including the “non-abelian” version of the affine Toda theory.

PROGRESS IN CLASSICALLY SOLVING TEN-DIMENSIONAL SUPERSYMMETRIC REDUCED YANG-MILLS THEORIES

Abstract It is shown that there exists an on-shell light cone gauge where half of the fermionic components of the super vector potential vanish, so that part of the superspace flatness conditions becomes linear. After reduction to (1 + 1) space-time dimensions, the general solution of this subset of equations is derived. The remaining non-linear equations are written in a form which is analogous to Yang equations, albeit with superderivatives involving sixteen fermionic coordinates. It is shown that this non-linear part may, nevertheless, be solved by methods similar to powerful techniques previously developed for the (purely bosonic) self-dual Yang Mills equations in four dimensions.

GAUGE CONDITIONS FOR THE CONSTRAINED-WZNW-TODA REDUCTIONS

Abstract There is a constrained-WZNW-Toda theory for any simple Lie algebra equipped with an integral gradation. It is explained how the different approaches to these dynamical systems are related by gauge transformations. Combining Gauss decompositions in relevant gauges, we unify formulae already derived, and explicitly determine the holomorphic expansion of the conformally reduced WZNW solutions — whose restriction gives the solutions of the Toda equations. The same takes place also for semi-integral gradations. Most of our conclusions are also applicable to the affine Toda theories.

W∞-geometry and associated continuous Toda system

Abstract We discuss an infinite-dimensional Kahlerian manifold associated with the area-preserving diffeomorphisms on a two-dimensional torus, and, correspondingly, with a continuous limit of the A r Toda system. In particular, a continuous limit of the A r Grassmannians and a related Plucker type formula are introduced as relevant notions for the W ∞ -geometry of the self-dual Einstein space with the rotational Killing vector.

Some explicit solutions of the Lamé and Bourlet type equations

Some special solutions to the multidimensional Lame and Bourlet type equations are constructed in an explicit form.