Solutions of bigraded Toda hierarchy

TL;DR

This paper explores the $(N,M)$-bigraded Toda hierarchy, revealing symmetries, deriving Hirota bilinear forms, and constructing explicit solutions including rational solutions expressed via Schur polynomials.

Contribution

It introduces the symmetry between $(N,M)$- and $(M,N)$-bigraded Toda hierarchies, derives their Hirota bilinear form, and constructs explicit solutions using moment matrices and Schur polynomials.

Findings

Established symmetry between $(N,M)$- and $(M,N)$-bigraded Toda hierarchies

Derived Hirota bilinear form for commuting flows

Constructed explicit solutions including rational solutions with Schur polynomials

Abstract

The $(N,M)$-bigraded Toda hierarchy is an extension of the original Toda lattice hierarchy. The pair of numbers $(N,M)$ represents the band structure of the Lax matrix which has $N$ upper and $M$ lower diagonals, and the original one is referred to as the $(1,1)$-bigraded Toda hierarchy. Because of this band structure, one can introduce $M+N-1$ commuting flows which give a parametrization of a small phase space for a topological field theory. In this paper, we first show that there exists a natural symmetry between the $(N,M)$- and $(M,N)$-bigraded Toda hierarchies. We then derive the Hirota bilinear form for those commuting flows, which consists of two-dimensional Toda hierarchy, the discrete KP hierarchy and its B\"acklund transformations. We also discuss the solution structure of the $(N,M)$-bigraded Toda equation in terms of the moment matrix defined via the wave operators associated with the Lax operator, and construct some of the explicit solutions. In particular, we give the rational solutions which are expressed by the products of the Schur polynomials corresponding to non-rectangular Young diagrams.