Solutions of bigraded Toda hierarchy

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 Bäcklund 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.

29 pages

References in corpus (8)

• The Extended Bigraded Toda hierarchy
• Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems
• Block type symmetry of bigraded Toda Hierarchy
• Tau function and Hirota bilinear equations for the Extended bigraded Toda Hierarchy
• Combinatorics of dispersionless integrable systems and universality in random matrix theory
• Iso-spectral deformations of general matrix and their reductions on Lie algebras
• Two extensions of 1D Toda hierarchy
• A class of integrable lattices and KP hierarchy

Cited by in corpus (15)

• Block type symmetry of bigraded Toda Hierarchy
• The extended $Z_N$-Toda hierarchy
• Dispersionless bigraded Toda Hierarchy and its additional symmetry
• Block algebra in two-component BKP and D type Drinfeld-Sokolov hierarchies
• On the extended multi-component Toda hierarchy
• Quantum Torus symmetry of the KP, KdV and BKP hierarchies
• Sato theory on the $q$-Toda hierarchy and its extension
• Quantum torus algebras and B(C) type Toda systems
• Multi-fold Darboux transformations of the extended bigraded Toda hierarchy
• Constrained lattice-field hierarchies and Toda system with Block symmetry
• Additional symmetry of the modified extended Toda hierarchy
• Bosonic symmetries of the extended fermionic $(2N,2M)$-Toda hierarchy
• A strongly coupled extended Toda hierarchy and its Virasoro symmetry
• Regular solution and lattice miura transformation of bigraded Toda Hierarchy
• Integrability on generalized $q$-Toda equation and hierarchy