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

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Guido Carlet
Johan van de Leur
  • Fonction : Auteur
Hessel Posthuma
  • Fonction : Auteur
Sergey Shadrin
  • Fonction : Auteur

Résumé

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.

Dates et versions

hal-03060389 , version 1 (14-12-2020)

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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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