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Register a new userHigher genera Catalan numbers and Hirota equations for extended nonlinear Schroedinger hierarchy
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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, Grothendiecks dessins denfants, 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 Schrodinger 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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The well-known Greens function method has been recently generalized to nonlinear second order differential equations. In this paper we study possibilities of exact Greens function solutions of nonlinear differential equations of higher order. We show that, if the nonlinear term satisfies a generalized homogeneity property, then the nonlinear Greens function can be represented in terms of the homogeneous solution. Specific examples and a numerical error analysis support the advantage of the method. We show how, for the Bousinesq and Kortweg-de Vries equations, we are forced to introduce higher order Green functions to obtain the solution to the inhomogeneous equation. The method proves to work also in this case supporting our generalization that yields a closed form solution to a large class of nonlinear differential equations, providing also a formula easily amenable to numerical evaluation.
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We study the behavior of the soliton solutions of the equation i((partial{psi})/(partialt))=-(1/(2m)){Delta}{psi}+(1/2)W_{{epsilon}}({psi})+V(x){psi} where W_{{epsilon}} is a suitable nonlinear term which is singular for {epsilon}=0. We use the strong nonlinearity to obtain results on existence, shape, stability and dynamics of the soliton. The main result of this paper (Theorem 1) shows that for {epsilon}to0 the orbit of our soliton approaches the orbit of a classical particle in a potential V(x).
The $q,t$-Catalan numbers can be defined using rational functions, geometry related to Hilbert schemes, symmetric functions, representation theory, Dyck paths, partition statistics, or Dyck words. After decades of intensive study, it was eventually proved that all these definitions are equivalent. In this paper, we study the similar situation for higher $q,t$-Catalan numbers, where the equivalence of the algebraic and combinatorial definitions is still conjectural. We compute the limits of sever
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