On the extended multi-component Toda hierarchy
Chuanzhong Li, Jingsong He

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.
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