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 Schrödinger 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.

我们考虑了第2级的Dubrovin–Frobenius流形，该流形的特殊扩展控制着加泰罗尼亚数的较高属一般性的枚举，或者等效地，对表面，带状图，Grothendieck的dessins d'enfants进行枚举，严格单调的Hurwitz数，或曲线的模空间中的晶格点。Liu，Zhang和Zhou猜想，该Dubrovin–Frobenius流形的全部分配函数是扩展的非线性Schrödinger层次结构的tau函数，是对Kadomtsev–Petviashvili层次结构的一种特殊合理化还原。我们证明了他们的猜想的一种版本，该猜想专门针对Givental–Milanov方法，该方法允许构造分区函数的Hirota二次方程，然后从中推导Lax表示。