The Extended Bigraded Toda hierarchy

TL;DR

This paper extends the Toda hierarchy by introducing additional variables, constructs new operators for a Lax pair, and explores its bihamiltonian structure, tau function, and dispersionless limit related to Frobenius manifolds.

Contribution

It introduces a generalized extended bigraded Toda hierarchy with new operators and a bihamiltonian framework, linking it to Frobenius manifolds and integrable systems.

Findings

Construction of roots and logarithms of the Lax operator.

Bihamiltonian formulation of the hierarchy.

Existence of a tau function for solutions.

Abstract

We generalize the Toda lattice hierarchy by considering N+M dependent variables. We construct roots and logarithms of the Lax operator which are uniquely defined operators with coefficients that are $\epsilon$-series of differential polynomials in the dependent variables, and we use them to provide a Lax pair definition of the extended bigraded Toda hierarchy. Using R-matrix theory we give the bihamiltonian formulation of this hierarchy and we prove the existence of a tau function for its solutions. Finally we study the dispersionless limit and its connection with a class of Frobenius manifolds on the orbit space of the extended affine Weyl groups of the $A$ series.