A polynomial bracket for the dubrovin–zhang hierarchies

Alexandr Buryak University of Amsterdam Hessel Posthuma University of Amsterdam Sergey Shadrin University of Amsterdam

Differential Geometry mathscidoc:1609.10282

Journal of Differential Geometry, 92, (1), 153-185, 2012
We define a hierarchy of Hamiltonian PDEs associated to an arbitrary tau-function in the semi-simple orbit of the Givental group action on genus expansions of Frobenius manifolds. We prove that the equations, the Hamiltonians, and the bracket are weightedhomogeneous polynomials in the derivatives of the dependent variables with respect to the space variable. In the particular case of a conformal (homogeneous) Frobenius structure, our hierarchy coincides with the Dubrovin–Zhang hierarchy that is canonically associated to the underlying Frobenius structure. Therefore, our approach allows to prove the polynomiality of the equations, Hamiltonians, and one of the Poisson brackets of these hierarchies, as conjectured by Dubrovin and Zhang.
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@inproceedings{alexandr2012a,
  title={A POLYNOMIAL BRACKET FOR THE DUBROVIN–ZHANG HIERARCHIES},
  author={Alexandr Buryak, Hessel Posthuma, and Sergey Shadrin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160913231603801036944},
  booktitle={Journal of Differential Geometry},
  volume={92},
  number={1},
  pages={153-185},
  year={2012},
}
Alexandr Buryak, Hessel Posthuma, and Sergey Shadrin. A POLYNOMIAL BRACKET FOR THE DUBROVIN–ZHANG HIERARCHIES. 2012. Vol. 92. In Journal of Differential Geometry. pp.153-185. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160913231603801036944.
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