On Discrete Symmetries of the Multi-Boson KP Hierarchies

We show that the multi-boson KP hierarchies possess a class of discrete symmetries linking them to the discrete Toda systems. These discrete symmetries are generated by the similarity transformation of the corresponding Lax operator. This establishes a canonical nature of the discrete transformations. The spectral equation, which defines both the lattice system and the corresponding Lax operator, plays a key role in determining pertinent symmetry structure. We also introduce a concept of the square-root lattice leading to a family of new pseudo-differential operators with covariance under additional B\"{a}cklund transformations.Comment: 11 pgs, LaTeX, IFT-P/75/93, UICHEP-TH/93-1

On Non-Linear W-Infinity Symmetry of Generalized Liouville and Conformal Toda Models

Invariance under non-linear ${\sf {\hat W}}_{\infty}$ algebra is shown for the two-boson Liouville type of model and its algebraic generalizations, the extended conformal Toda models. The realization of the corresponding generators in terms of two boson currents within KP hierarchy is presented.Comment: 10 pgs, LaTeX, IFT-P.038/9

Manifestly Supersymmetric Lax Integrable Hierarchies

A systematic method of constructing manifestly supersymmetric $1+1$-dimensional KP Lax hierarchies is presented. Closed expressions for the Lax operators in terms of superfield eigenfunctions are obtained. All hierarchy equations being eigenfunction equations are shown to be automatically invariant under the (extended) supersymmetry. The supersymmetric Lax models existing in the literature are found to be contained (up to a gauge equivalence) in our formalism.Comment: LaTeX, 10 pg

Compatible Poisson Structures of Toda Type Discrete Hierarchy

An algebra isomorphism between algebras of matrices and difference operators is used to investigate the discrete integrable hierarchy. We find local and non-local families of R-matrix solutions to the modified Yang-Baxter equation. The three R-theoretic Poisson structures and the Suris quadratic bracket are derived. The resulting family of bi-Poisson structures include a seminal discrete bi-Poisson structure of Kupershmidt at a special value.Comment: 22 pages, LaTeX, v3: Minor change

Darboux-Egoroff Metrics, Rational Landau-Ginzburg Potentials and the Painleve VI Equation

We present a class of three-dimensional integrable structures associated with the Darboux-Egoroff metric and classical Euler equations of free rotations of a rigid body. They are obtained as canonical structures of rational Landau-Ginzburg potentials and provide solutions to the Painleve VI equation.Comment: 20 page

Construction of KP Hierarchies in Terms of Finite Number of Fields and their Abelianization

The $2M$-boson representations of KP hierarchy are constructed in terms of $M$ mutually independent two-boson KP representations for arbitrary number $M$. Our construction establishes the multi-boson representations of KP hierarchy as consistent Poisson reductions of standard KP hierarchy within the $R$-matrix scheme. As a byproduct we obtain a complete description of any finitely-many-field formulation of KP hierarchy in terms of Darboux coordinates with respect to the first Hamiltonian structure. This results in a series of representations of \Win1\, algebra made out of arbitrary even number of boson fields.Comment: 12 p., LaTeX, minor typos corrected, BGU-93/2/June-P