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Higher genera Catalan numbers and Hirota equations for extended nonlinear Schroedinger hierarchy

Abstract : We consider the Dubrovin--Frobenius manifold of rank $2$ whose genus expansion at a special point controls the enumeration of a higher genera generalization of the Catalan numbers, or, equivalently, the enumeration of maps on surfaces, ribbon graphs, Grothendieck's dessins d'enfants, strictly monotone Hurwitz numbers, or lattice points in the moduli spaces of curves. Liu, Zhang, and Zhou conjectured that the full partition function of this Dubrovin--Frobenius manifold is a tau-function of the extended nonlinear Schr\"odinger hierarchy, an extension of a particular rational reduction of the Kadomtsev--Petviashvili hierarchy. We prove a version of their conjecture specializing the Givental--Milanov method that allows to construct the Hirota quadratic equations for the partition function, and then deriving from them the Lax representation.
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https://hal-univ-bourgogne.archives-ouvertes.fr/hal-03060389
Contributeur : Guido Carlet <>
Soumis le : lundi 14 décembre 2020 - 00:32:57
Dernière modification le : jeudi 28 janvier 2021 - 10:28:03

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  • HAL Id : hal-03060389, version 1
  • ARXIV : 2012.03239

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Guido Carlet, Johan van de Leur, Hessel Posthuma, Sergey Shadrin. Higher genera Catalan numbers and Hirota equations for extended nonlinear Schroedinger hierarchy. 2020. ⟨hal-03060389⟩

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