Letters in Mathematical Physics

, Volume 108, Issue 2, pp 261–283 | Cite as

Fractional Volterra hierarchy

  • Si-Qi Liu
  • Youjin Zhang
  • Chunhui Zhou


The generating function of cubic Hodge integrals satisfying the local Calabi–Yau condition is conjectured to be a tau function of a new integrable system which can be regarded as a fractional generalization of the Volterra lattice hierarchy, so we name it the fractional Volterra hierarchy. In this paper, we give the definition of this integrable hierarchy in terms of Lax pair and Hamiltonian formalisms, construct its tau functions, and present its multi-soliton solutions.


Fractional Volterra hierarchy Hodge integral Hamiltonian structure Tau structure Darboux transformation 

Mathematics Subject Classification

37K10 53D45 



We are grateful to Boris Dubrovin and Di Yang for sharing with us their discovery of the relation of the special cubic Hodge integrals with Eq. (1.8) and for helpful discussions. We would also like to thank Fedor Petrov and Vladimir Dotsenko for their proof of the four identities given in the end of Sect. 3, and the referee for the suggestion to simplify some proofs of the paper. This work is supported by NSFC No. 11371214 and No. 11471182.


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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingPeople’s Republic of China

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