Fractional Volterra hierarchy
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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.
KeywordsFractional Volterra hierarchy Hodge integral Hamiltonian structure Tau structure Darboux transformation
Mathematics Subject Classification37K10 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.
- 4.Dubrovin, B., Yang, D.: Private communications (2016)Google Scholar
- 6.Dubrovin, B., Liu, S.-Q., Yang, D., Zhang, Y.: Hodge-GUE correspondence and the discrete KdV equation, arXiv:1612.02333 [math-ph]
- 8.Gu, C., Hu, H., Zhou, Z.: Darboux Transformations in Integrable Systems. Theory and Their Applications to Geometry. Mathematical Physics Studies, vol. 26. Springer, Dordrecht (2005)Google Scholar
- 12.Kupershmidt, B.A.: Discrete Lax equations and differential-difference calculus. Astérisque No. 123 (1985)Google Scholar
- 14.Liu, S.-Q., Petrov, F., Dotsenko, V.: Two combinatorial identities. Math Overflow. http://mathoverflow.net/questions/261414
- 15.Mariño, M., Vafa, C.: Framed knots at large N. In: Adam, A., Morava, J., Ruan, Y. (eds) Orbifolds in Mathematics and Physics: Proceedings of a Conference on Mathematical Aspects of Orbifold String Theory May 4–8, 2001, University of Wisconsin, Madison, WI. Contemporary Mathematics, vol. 310, pp. 185–204. American Mathematical Society, Providence, RI (2002)Google Scholar
- 17.Mumford, D.: Towards an enumerative geometry of the moduli space of curves. In: Artin, M., Tate, J. (eds) Arithmetic and Geometry. Progress in Mathematics, vol. 36, pp. 271–328. Birkhäuser, Boston, MA (1983)Google Scholar
- 20.Ueno, K., Takasaki, K.: Toda lattice hierarchy. In: Okamoto, K. (ed) Group Representations and Systems of Differential Equations. Advanced Studies in Pure Mathematics, vol. 4. North-Holland/Kinokuniya, Amsterdam (1984)Google Scholar