We consider the Hankel determinant formula of the τ functions of the Toda equation. We present a relationship between the determinant formula and the auxiliary linear problem, which is characterized by a compact formula for the τ functions in the framework of the KP theory. Similar phenomena that have been observed for the Painlevé II and IV equations are recovered. The case of finite lattice i...

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...

Based on the recent formulation of a general scheme to construct boundary Lax pairs, we develop this systematic construction for the A (1) n affine Toda field theories (ATFT). We work out explicitly the first two models of the hierarchy, i.e. the sineGordon (A (1) 1 ) and the A (1) 2 models. The A (1) 2 Toda theory is the first non-trivial example of the hierarchy that exhibits two distinct typ...

The Banach Poisson geometry of multi-diagonal Hamiltonian systems having infinitely many integrals in involution is studied. It is shown that these systems can be considered as generalizing the semi-infinite Toda lattice which is an example of a bidiagonal system, a case to which special attention is given. The generic coadjoint orbits of the Banach Lie group of bidiagonal bounded operators are...