2012
DOI: 10.1016/j.geomphys.2012.03.006
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On deformations of quasi-Miura transformations and the Dubrovin–Zhang bracket

Abstract: Abstract. In our recent paper we proved the polynomiality of a Poisson bracket for a class of infinite-dimensional Hamiltonian systems of PDE's associated to semi-simple Frobenius structures. In the conformal (homogeneous) case, these systems are exactly the hierarchies of Dubrovin-Zhang, and the bracket is the first Poisson structure of their hierarchy.Our approach was based on a very involved computation of a deformation formula for the bracket with respect to the Givental-Y.-P. Lee Lie algebra action. In th… Show more

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Cited by 45 publications
(77 citation statements)
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“…They have recently confirmed that such an integrable hierarchy exists, provided that a certain conjecture about polynomiality of the Poisson brackets holds [31]. This conjecture was partially proved by Buryak-Posthuma-Shadrin [15,16] (the polynomiality of the second bracket is still an open problem). Another approach is to derive Hirota's bilinear equations for the taufunction; see [91,87,88,89,90,58,60,45].…”
Section: N I=1mentioning
confidence: 87%
“…They have recently confirmed that such an integrable hierarchy exists, provided that a certain conjecture about polynomiality of the Poisson brackets holds [31]. This conjecture was partially proved by Buryak-Posthuma-Shadrin [15,16] (the polynomiality of the second bracket is still an open problem). Another approach is to derive Hirota's bilinear equations for the taufunction; see [91,87,88,89,90,58,60,45].…”
Section: N I=1mentioning
confidence: 87%
“…The theory was later generalized in [BPS12b]. In this section we follow the approach from [BPS12b] (see also [BPS12a]). …”
Section: Dubrovin-zhang Hierarchy For Cpmentioning
confidence: 99%
“…In [14] the above mentioned Hamiltonian structure of the deformed hierarchy is obtained from the first Hamiltonian structure of the principal hierarchy via the quasi-Miura transformation (6.5). A proof of the polynomiality of the topological deformation of the principal hierarchy and of the Hamiltonian operator P 1 are given in [26] and [27]. For the second Hamiltonian structure of the principal hierarchy, the following conjecture is given in [14].…”
Section: The Topological Deformationsmentioning
confidence: 99%