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Higher genera Catalan numbers and Hirota equations for extended nonlinear Schrödinger hierarchy. (English) Zbl 1468.37051

Authors’ 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. S.-Q. Liu et al. [J. Geom. Phys. 97, 177–189 (2015; Zbl 1319.37045)] 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.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
14H81 Relationships between algebraic curves and physics
14H70 Relationships between algebraic curves and integrable systems

Citations:

Zbl 1319.37045
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