On the extended multi-component Toda hierarchy

TL;DR

This paper introduces the extended multi-component Toda hierarchy, providing its flow equations, Hirota bilinear form, tau functions, Darboux transformation, and bi-Hamiltonian structure, with potential applications in topological field and Gromov-Witten theories.

Contribution

It constructs the extended multi-component Toda hierarchy with new bilinear equations, tau functions, and Hamiltonian structures, advancing integrable systems theory.

Findings

Derived the extended flow equations for the hierarchy

Established Hirota bilinear equations and tau functions

Presented Darboux transformation and bi-Hamiltonian structure

Abstract

The extended flow equations of the multi-component Toda hierarchy are constructed. We give the Hirota bilinear equations and tau function of this new extended multi-component Toda hierarchy(EMTH). Because of logarithmic terms, some extended vertex operators are constructed in generalized Hirota bilinear equations which might be useful in topological field theory and Gromov-Witten theory. Meanwhile the Darboux transformation and bi-Hamiltonian structure of this hierarchy are given. From the Hamiltonian tau symmetry, we give another different tau function of this hierarchy with some unknown mysterious connections with the one defined from the point of wave functions.