Recursion relations for orthogonal polynominals, arising in the study of the one-matrix model of two-dimensional gravity, are shown to be equvalent to the equations of the Todachain hierarchy supplemented by additional Virasoro constraints. This is the case without the double scaling limit. A discrete time variable to the matrix model is introduced. The discrete time dependent partition functio...

The Eguchi–Hanson, Taub–NUT and Atiyah–Hitchin metrics are the only complete non–singular SO(3)–invariant hyper–Kahler metrics in four dimensions. The presence of a rotational SO(2) isometry allows for their unified treatment based on solutions of the 3–dim continual Toda equation. We determine the Toda potential in each case and examine the free field realization of the corresponding solutions...

For a family of Poisson algebras, parametrized by r ∈ Z, and an associated Lie algebraic splitting, we consider the factorization of given canonical transformations. In this context we rederive the recently found r-th dispersionless modified KP hierachies and r-th dispersionless Dym hierarchies, giving a new Miura map among them. We also found a new integrable hierarchy which we call the r-th d...

The low energy effective actions which arise from string theory or M-theory are considered in the cosmological context, where the graviton, dilaton and antisymmetric tensor field strengths depend only on time. We show that previous results can be extended to include cosmological solutions that are related to the EN Toda equations. The solutions of the Wheeler-DeWitt equation in minisuperspace a...

— Rational solutions of the second, third and fourth Painlevé equations (PII–PIV) can be expressed in terms of logarithmic derivatives of special polynomials that are defined through coupled second order, bilinear differential-difference equations which are equivalent to the Toda equation. In this paper the structure of the roots of these special polynomials, and the special polynomials associa...